Catalan-many tropical morphisms to trees; Part I: Constructions
Jan Draisma Alejandro Vargas
  		
	   
		We investigate the tree gonality of a
		genus-g metric graph, defined as the minimum
		degree of a tropical morphism from any
		tropical modification of the metric graph to a
		metric tree. We give a combinatorial
		constructive proof that this number is at most
		⌈ g/2 ⌉ + 1, a fact whose proofs
		so far required an algebro-geometric detour
		via special divisors on curves. For even
		genus, the tropical morphism which realizes the
		bound belongs to a family of tropical
		morphisms that is pure of dimension 3g-3 and
		that has a generically finite-to-one map onto
		the moduli space of genus-g metric graphs. Our methods focus on the study of such families. 
		This is part I in a series of two
		papers: in part I we fix the
		combinatorial type of the metric graph to show a bound on tree-gonality, while
		in part II we vary the combinatorial type and
		show that the number of tropical morphisms,
		counted with suitable multiplicities, is the
		same Catalan number that counts morphisms
		from a general genus-g curve to the projective line.
	
	
	
	On extrema of the objective functional for short-time generation of single qubit quantum gates
    Alexander Pechen^1[Corresponding author. E-mail: apechen@gmail.comapechen@gmail.com; Webpage: http://www.mathnet.ru/eng/person17991mathnet.ru/eng/person17991]   and Nikolay Il'in^2
    
========================================================================================================================================================================================



 §.§ Introduction
 
	We investigate the tree gonality of metric graphs. This concept has its roots in algebraic geometry. The gonality of a genus-g curve X is the minimum degree of a non-constant morphism from X to ℙ^1. The non-constant morphisms ϕ: X →ℙ^1 are in one-to-one correspondence with linear series of degree d and dimension 1 on X, so equivalently the gonality of X is the minimum degree of such linear series. Brill-Noether theory for algebraic curves establishes that the gonality of X is at most ⌈ g/2 ⌉ + 1. The set of curves for which this bound is sharp is dense in the moduli space of genus-g curves ℳ_g.
	
	
	A striking interplay between
	graphs and algebraic curves has been discovered over the last two
	decades. In tropical geometry the objects that play the role of algebraic curves and of the projective line are metric graphs and metric trees, respectively. This goes beyond a
	formal analogy of concepts: specialization
	lemmas and correspondence theorems between both
	settings are an active field of research at the
	interface between tropical and non-archimedean geometry 
	(see, e.g., <cit.>). 
	
	There are two definitions of gonality for (metric) graphs which are relevant to our story.
	A chip-firing game on a (metric) graph yields a divisor theory (see
	<cit.>). The divisorial gonality of a (metric) graph is the minimum degree of a rank-1 divisor. 
	On the other hand, certain maps between metric graphs with suitable balancing
	conditions bahave similarly to morphisms between
	curves (see <cit.>). We call these maps tropical morphisms. The
	tree gonality of a metric graph is the minimum
	degree of a tropical morphism from any
	tropical modification of the metric graph to a
	metric tree. Tropical modification is an operation 
	which retracts or attaches edges ending in a valency-1 vertex, and is useful to satisfy balancing conditions. 
	
	Divisorial gonality mimics the definition of gonality of X via linear series, while tree gonality mimics the one via non-constant morphisms. Unlike the algebro-geometric setting, in the tropical world divisorial gonality and tree-gonality do not coincide. If : → is a tropical morphism to a tree, then the divisor ∑_x' ∈(x)  m_(x') x' has rank at least 1 for any choice of x in  (see Subsection <ref> for notation). So tree gonality is an upper bound of divisorial gonality. On the other hand, there are families of graphs with divisorial gonality equal to 2, and tree gonality equal to ⌈ g/2 ⌉ + 1, where g is the first Betti number of the graph. At the time of writing, perhaps the most famous open question on divisorial gonality is the following:
	
	
	
	
	 
		The divisorial gonality of a finite connected graph G is at most ⌈ g/2  ⌉ + 1, where g is the first Betti number of G.
	
	
	This conjecture has attracted a sizeable amount of attention from the community. If one replaces G with a metric graph , then the result is proven in <cit.>. The proof requires an algebro-geometric detour via special divisors on curves. There is wide interest for a purely combinatorial proof, as it is hoped it may yield methods better suited to tackle Conjecture <ref>.
	Our main contribution is such a purely combinatorial proof:
	
	 
		The tree gonality of a genus-g metric graph  is at most ⌈ g/2⌉  + 1.
	
	
	In fact, we manage to construct tropical morphisms that witness the bound ⌈ g/2⌉ + 1 on tree-gonality, so beyond being combinatorial, the proof is also effective. There is some computer code and visualizations available at <cit.>.
	
	
	
	
	
	
	
	
	
	

 §.§ Proof idea
 The main idea behind our proof is short, elegant, and ubiquitous: we first analyse a favourable case and then continuously deform the result. We need an initial family of graphs  for which we have a construction of a tropical morphism : → attaining the gonality bound; and a procedure to deform  into : → for any given  via a deformation path in g, the moduli space of metric graphs. We argue that for this strategy to work we need that , and all but finitely many of the points in the deformation path to , belong to cones of tropical morphisms that we call full-dimensional.  Intuitively, these morphisms depend on the right number of parameters to cover g. 
	
	We use this idea for both parts of this series of articles, only the ingredients change. In Part I we use constructions from <cit.> as initial families, which gives us a tropical morphism for each possible trivalent graph H (this H is the combinatorial type, defined in Subsection <ref>). Thus, the deformation procedure only has to handle paths where the combinatorial type is the same throughout the deformation path in g. Setting up the deformation requires analysing 16 cases and a balancing relation between certain determinants associated to
	the tropical morphisms. This balancing relation, closely
	related to the theory of morphisms between tropical fans
	<cit.>, ensures
	that we can always proceed further in our desired path from
	 to . 
	
	In Part II the initial point of the deformation is our own family of graphs, which we call caterpillars of loops. We understand them better than the constructions of <cit.> in that we are able to prove that if  is a caterpillar of loops, then there are exactly Catalan-many tropical morphisms realizing the gonality bound. We also strengthen the deformation procedure. By analyzing 8 further cases we can deform past changes of combinatorial type. We show that with suitable multiplicity the count of tropical morphisms realizing the gonality bound remains constant thorough deformation. 
	
	To carry out this simple idea we introduce a technical framework to describe what is a full-dimensional cone and its properties, set up the deformation procedure, and support the case work required to prove the correctness of deformation. The road through Parts I and II is long, but it comes with two rewards: we achieve a combinatorial proof of the gonality bound; at the finish line we arrive at a tropical space that parametrizes tropical morphisms attaining the gonality bound, which surjects onto g, is connected through codimension 1, and has fibres with Catalan-many points. This mirrors the beautiful classical story due Eisenbud and Harris <cit.>. It would be interesting to understand how
	our work relates to the tropicalization of the
	space of admissible covers from <cit.>, but this
	is beyond the scope of these combinatorial papers. 
	
	

 §.§ Organization of article

	Section <ref> recalls several concepts on
	graphs, metric graphs, and g.
	Section <ref> develops a formalism for
	tropical morphisms that highlights the combinatorics. Here we introduce full-dimensional cones of tropical morphisms.
	Section <ref> introduces the main
	combinatorial gadget of this article, the gluing datum. We use it to study the properties of full-dimensional cones. Section <ref> is a brief
	pause from developing the theory, with a fully worked out
	example suggesting the deformation procedure—in fact, we
	encourage the reader to go through this example before going
	through the more technical parts of the article! Section <ref> resumes the theory with a study of the deformation operation, and a proof of the main result. This proof depends on many constructions that we carry out in Section <ref>.
	
	
	
	

§ ACKNOWLEDGEMENTS
 
	We thank Erwan Brugallé, Melody Chan, Hannah Markwig, and Sam Payne for enlightening discussions, pointers to literature, and insights into future research directions. We thank the hospitality and the marvellous tropical environment at the Mittag-Leffler institute during the semester on “Tropical Geometry, Amoebas and Polytopes”, where part of this research was done. We thank the general and specific remarks of three anonymous referees, which helped us improve greatly the exposition of the paper.
	
	

§ BACKGROUND
 
	
	

 §.§ Graphs
 
	A graph G is a pair (V(G),E(G)) of disjoint sets, the vertex and edge sets respectively, and a map ι_G defined on E(G) encoding incidences,  whose values are multisets of two elements of V(G). 
	Given an edge e with ι_G(e) = {A, B} we call A, B the ends of e, and say that A and e are incident. A loop is an edge e such that its two ends are equal. A subgraph of G is a graph G' such that V(G') ⊂ V(G), E(G') ⊂ E(G), and ι_G' equals the restriction ι_G|_E(G'). A subset S of V(G) induces a subgraph by taking for edge set all the edges of G with both ends in S. Likewise, a subset S of E(G) induces a subgraph by taking for vertex set the ends of all the edges in S. A graph G is finite if both V(G) and E(G) are finite sets; and loopless if it has no loops.
	
	A path of G is a sequence ⟨ A_0, e_1, A_1, …, e_μ, A_μ⟩ of alternately vertices and edges, where no element is repeated, such that consecutive elements are incident and μ≥ 1. The ends of a path are the vertices A_0, A_μ. The remaining vertices A_1, …, A_μ - 1 are interior.  We call μ the length of P. We write x ∈ P to mean that x is an element in the sequence P. If P is a path with ends A, B, and e in E(G) ∖ P is an edge that also has ends A, B, then we call the subgraph consisting of e together with P a cycle. Given a vertex A, its connected component is the subgraph induced by the set of ends of all paths with A ∈ P.  The vertex set V(G) is partitioned into connected components. A graph is connected if it has a single connected component.
	
	Let G be a finite, connected graph. Set g(G)=|E(G)|-|V(G)|+1. Following a convention due to <cit.>, we call this number the genus of G. 
	If G has genus 0, then we call G a tree.
	Write x ∈ G for x ∈ V(G) ∪ E(G). Let A be a vertex. Write G A for the subset of E(G) incident to A. The valency G A of A is the number of edges in G A, with loops counting twice. The minimum valency of G is the number min_A ∈ V(G) G A. We write A and A when G is clear from the context. A vertex is monovalent, divalent, trivalent, or n-valent if its valency equals 1, 2, 3, or n, respectively. We call a monovalent vertex of a tree a leaf. 
	
	Let S ⊂ E(G). The deletion of S, denoted by G ∖ S, is the graph with edge set E(G) ∖ S and incidence map ι_G|_E(G) ∖ S. The contraction of S, denoted by G / S, is the graph (V(G)/∼, E(G) ∖ S), 
	where ∼ identifies the vertices A and B if they are the ends of some e in S,  with incidence map π∘ι_G|_E(G) ∖ S with π : V(G) → V(G)/ ∼ the canonical projection.  
	The subdivision of an edge e yields the graph (V(G) ∪{C}, (E(G) ∖{e}) ∪{e_1, e_2}), where the ends of e_1 are A, C, and of e_2 are C, B.
	
	Let G, G' be two graphs. A graph morphism is a map γ: V(G) ∪ E(G) → V(G') ∪ E(G') such that: γ(V(G)) ⊆ V(G'); if γ(e) ∈ V(G') for an edge e in E(G) with ends A and B, then γ(e) = γ(A) = γ(B); if γ(e) ∈ E(G') for an edge e in E(G) with ends A and B, then the ends of γ(e) are γ(A) and γ(B). In essence, a morphism is an incidence preserving map that can contract edges. If γ(E(G)) ⊆ E(G'), then we call γ a homomorphism. An isomorphism is a bijective homomorphism; its inverse is then a homomorphism as well.
	
	An edge (resp. vertex) labelling is an injective map λ from E(G) (resp. V(G)) to a set S. A total order ≤_S on S induces a total order on E(G) by letting e ≤_E(G) e' when λ(e) ≤_S λ(e'). We use ℕ as labelling set, with its natural order. In the next section we consider elements from the vector space ^E(G) (real valued functions on E(G)). By choosing an edge labelling λ this space is identified with ^|E(G)|: we write e_j ∈ E(G) for λ(j) (we use this notation whenever a λ has been chosen); we take as ordered basis of ℝ^E(G) the functions ℓ_i that map e_j to 1 if i = j, and zero otherwise.
	
	

 §.§ Metric graphs
  A length function for G is a map ℓ:E(G) →. The pair (G, ℓ) gives rise to a one-dimensional CW-complex  by gluing the disjoint union of closed real intervals _e =[0,ℓ(e)], one for each e in E(G), along their endpoints in the manner prescribed by G.  
	We call the first Betti number of the CW-complex  the genus of , denoted by g(). It coincides with g(G).
	This CW-complex is equipped with the shortest-path metric in which the intervals _e have lengths ℓ(e). This metric space is called a metric graph. We call G a model, and the pair (G, ℓ) a realization, of .
	
	Let = (G, ℓ_G). There are infinitely many realizations of . For example, we often consider the model G' obtained by subdividing e in E(G) into e_1, e_2, and taking as length function some ℓ' with ℓ'(e_1) + ℓ'(e_2) = ℓ(e), so (G', ℓ') also realizes . A vertex set of  is a finite subset S ⊂ of points such that ∖ S is homeomorphic to a disjoint union of open real intervals. Every realization (G, ℓ_G) of  induces a vertex set: the set of endpoints of the intervals _e. We refer to the intervals _e as the edges of .  Every vertex set S induces a realization (G_S, ℓ_S) of : the vertices of G_S are the points in S; the edges are the intervals ∖ S; the length function ℓ_S is given by the lengths of ∖ S. Thus, a choice of model G for  amounts to a choice of vertex set, and therefore edge set as well.
	
	
	
	
	
	
	
	We single out a particular realization for . 
	Consider points x in  such that for all ε > 0 the open ball B(x,ε) with centre x and radius ε is not isometric to the interval (-ε, ε). We call these essential vertices. The naming is due to the fact that if x is an essential vertex, then x cannot be in the topological interior of an edge _e. Thus, the set of essential vertices is contained in the set of endpoints of edges _e, and so there are finitely many. On the other hand, we call the metric graphs with an empty set of essential vertices metric loops. They have genus 1 and admit the graph with one vertex and one loop as a model.
	
	 
		Let  be a metric graph distinct from a metric loop, ℰ the essential vertices of , and S ⊂ a finite set of points. Then S is a vertex set if and only if ℰ⊂ S.
	
	
	The valency of a point x in  is the valency of x in G_S for S a vertex set with x in S.
	If  is not a metric loop, then we call the realization (G_ℰ, ℓ_ℰ) the essential realization. Thus, by Lemma <ref> the essential model G_ℰ is minimal in the sense that every other model arises from a sequence of edge subdivisions of G_ℰ. Hence, all the realizations are refinements of the essential realisation. This shows x to be well defined.
	
	
	
	

 §.§ Tropical modification
 
	In the literature, tropical modification of a metric graph = (G, ℓ) is an operation that deletes or attaches a segment ending in a monovalent point. For investigating tropical morphisms we consider metric graphs to be equivalent under a sequence of these operations. This allows to meet certain balancing conditions, and does not alter  in a significant manner. Our approach to tropical modification is based on the notion of dangling elements in the model G:
	
	
	 
		An edge e of G is dangling if deleting e produces two connected components and one of those is a tree. A vertex A of G is dangling if all e in E(A) are dangling.
	
	
	This definition detects points contained in a segment that has been attached to . Fix a model G for . Let x be in . If x is in the vertex set induced by G, then we say that x is dangling when its corresponding vertex in G is dangling. Otherwise x is in the interior of an edge _e and we say that x is dangling if e is dangling. It is straighforward to see that this definition is independent of the chosen realization (G, ℓ).
	
	Let  be the graph obtained by deleting all the dangling elements of G. By Definition <ref> we have that g(G) = g(). 
	Let = (, ℓ|_), where ℓ|_ is the restriction of ℓ to E(). We identify G with  and  with  under tropical modification. 
	Let H be the essential model of . The construction of  and Lemma <ref> together imply that:
	 
		Let  be a metric graph of genus at least 2, and ' a tropical modification of , such that ' has a model G' with minimum valency at least three. Then ' is isometric to  and G' is isomorphic to H.  
	
	
	Thus,  is a canonical representative of the equivalence class under isomorphism and tropical modification of . We call a graph with minimum valency at least three a combinatorial type. This is an important invariant under tropical modification, so we say that H is the combinatorial type of . 
	
	
	
	
	
	
	

 §.§ Moduli space of metric graphs
  
	We give a barebones account of the moduli space of genus-g metric graphs g.  If all the vertices of a graph are trivalent, we call it a trivalent combinatorial type, or simply trivalent. We denote by 𝔾_g the family of genus-g trivalent combinatorial types. For the construction of g we use the following straightforward lemma:
	
	 
		Let g ≥ 2 and H_0 be a genus-g combinatorial type. There exists H in 𝔾_g such that there is a sequence of edge contractions of H that yield a graph isomorphic to H_0.
	
	
	Given a graph G, we denote by C_G the set of length functions defined on G. This set has a cone structure, as it equals the positive orthant ^E(G). We identify each point ℓ of C_G with the metric graph (G, ℓ) to obtain the (rational polyhedral) cone C_G of metric graphs with model G.
	
	Motivated by Lemma <ref> we add some of the boundary points of C_G with the following convention: 
	given a map ℓ: E(G) →, contract all the edges of G for which ℓ(e) = 0, to get G';  
	the pair (G, ℓ) refers to the metric graph (G', ℓ|_G'). Let C_G ⊂^E(G) be the set of maps ℓ for which the metric graph (G, ℓ) has genus g, that is, all cycles of G have positive length. gluing together these completed cones gives rise to a space where points bijectively correspond to classes under tropical modification of genus-g metric graphs:
	
	The moduli space of genus-g metric graphs is the space
		
    g = (_H ∈𝔾_gC_H )  / ≅,

		where ≅ identifies points ℓ_H and ℓ_H' for which (H, ℓ_H) and (H', ℓ_H') are isometric. It is given the quotient topology. 
	
	
	Given a metric graph , by Lemma <ref> and <ref> there is a point in g corresponding to a graph  that is equivalent to  under tropical modification. By construction this point is unique.
	
	
		The reader might find in the literature a different use for the symbol g, albeit for constructions related to ours. They differ by imposing additional structure on the metric graph: marked points, legs of infinite length, weights on the vertices, and more. These are not required for our purposes, as they typically arise to encode more algebro-geometric information in a graph. See <cit.> and <cit.> for in-depth studies.
	
	
	
	
	
		Identifying isometric points complicates the geometry of g. We do not obtain a polyhedral cone complex, as one might expect. This issue is addressed in <cit.> with the introduction of abstract polyhedral cone complexes. See their Section 4.3 for a proof that g is an abstract polyhedral cone complex.
	
	
	Let H be a combinatorial type. From 2|E(H)| = ∑_A ∈ V(H) A one can conclude that H is trivalent if and only if |E(H)| = 3g-3. Hence, g is pure of dimension 3g-3.
	
	
	
	
	
	

 §.§ Notation
  
	In the next four sections the following letters, and small variations thereof, are used to denote specific objects. Metric graphs:  for metric graphs with g() ≥ 2;  for metric trees (due to tree being δέντρο in Greek), 
	that is genus 0 metric graphs; and  for the deletion of the dangling elements of . Finite graphs: G for a finite connected graph with g(G) ≥ 2,  A for its vertices and e for its edges; T for a finite tree, v for its vertices and t for its edges; and H for the combinatorial type of G. Morphisms:  for maps between finite graphs; and  for maps between metric graphs. We write [d] for the set  {1, 2, …, d }, and x ∈ G for x ∈ V(G) ∪ E(G).
	
	
	
	
	
	
	
	
	
	

§ TROPICAL MORPHISMS
  
	
	
	
	
	

 §.§ 
 
	
	We begin with a map encapsulating the combinatorial information of tropical morphisms. These are a special class of graph morphisms, augmented with an index map. 
	
	[discrete tropical morphism] 
		Let G, G' be loopless graphs, : G → G' a morphism, and m_: G → a map.
		 
			
  * m_ is an index map for  if for every e in E(G) we have that (e) ∈ V(G') if and only if m_(e) = 0. That is, only index zero edges are contracted by .
			
  *  is harmonic with index map m_ if it satisfies the balancing condition: for every A in V(G) and e' incident to (A) we have that
			
    m_(A) = ∑_e ∈ E(A) ∩φ(e') m_φ(e).

			Note that this makes the sum independent of the choice of e'.
			
  *  is non-degenerate if m_(x) ≥ 1 for all x in G.  
			
  * φ satisfies the Riemann-Hurwitz condition (RH-condition) if for every A in V(G):
			
    r_(A) = ( A - 2) - (φ(A) - 2) · m_φ (A) ≥ 0.

			
  * A discrete tropical morphism is a pair (, m_) consisting of a non-degenerate harmonic morphism φ with index map m_ that satisfies the RH-condition. We write to shorten. 
		
	
	
	The non-degeneracy condition ensures that no edge is collapsed. 
	
	
	 
		Definition <ref> is similar to Definition 2.1 of <cit.>, but there are differences:  condition (c) of Definition 2.1 in the cited work is required only for vertices of G (see discussion in Section 1.3 of <cit.>); an alternative formula for r_ is given on  Equation (10) of the cited work (see Lemma <ref>); and we do not have a vertex labelling representing extra genus. 
	 
	
	
	[formula for r_φ]  
		Let  be a with index map m_. Then
		
    r_(A) = 2(m_(A) - 1) - ∑_e ∈ E(A) (m_(e) - 1).

	
	
	
	
		By the balancing condition m_(A) ·(A) = ∑_e ∈ A m_(e). Thus,
		
    r_φ(A)    =  A - 2 - m_φ (A) · ((A) - 2)
       = 2(m_(A) - 1) +  A - ∑_e ∈ A m_(e)
       = 2(m_(A) - 1) - ∑_e ∈ A (m_(e) - 1).

		
	
	
	We always denote the index map of  by m_. So we write “ is a ” instead of “φ is a with index map m_”. We call G the source, and G' the target, of . The index m_ can be understood as a local degree: for A in V(G) restricting  to the subgraph G_A induced by E(A) gives a onto the subgraph induced by (E(A)); any fibre of |_ G_A has m_(A) elements, each element counted with multiplicity m_. Since G is connected this local degree gives rise to a global degree, where the constant obtained in the count of preimages with multiplicities is called the degree of : 
	
    = ∑_(A) = A' m_(A);

	this is independent of A' by the connectedness of G (see proof of Lemma 2.4 of <cit.>).
	
	 	
		Figure <ref> illustrates two _A, _B of degrees 3 and 4, respectively. The targets are drawn below, the sources above, and the fibres with dotted lines. The edges have index 1, unless noted otherwise with a number above the edge.
		

		
		0.95
			[t]0.6
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					(66,47) 0.62
				 
			
			[b].35 
				
				[scale=1.1]/278.pdf    
					(46,47) 0.62
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			[t].6 
				
				_A			
			
			[t].35 
				
				_B
			
			
			figure 
		
	
	
	
	This setup is parallel to the classical theory of maps between Riemann surfaces. There is a Riemann-Hurwitz formula relating the genera of the source and the target.
	
	
	[Riemann-Hurwitz formula]  
		Let : G → G' be a . Then
		
    2g(G) - 2 = · (2g(G')-2) + ∑_A ∈ V(G) r_(A).
 
	
	
	
	
		Recall that in a graph the sum of all the valencies is twice the number of edges:
		
    ∑_A ∈ V(G) r_(A)    = ∑_A ∈ V(G) ( A - 2 - m_φ (A) · ((A) - 2) )
       = 2|E(G)|-2|V(G)|- · (2|E(G')|-2|V(G')|)
       = 2g(G) - 2 - · (2· g(G')-2).

		Rearrange terms to obtain the result.
	
	The RH-condition implies that the sum ∑_A ∈ V(G) r_(A) is positive, thus g(G) ≥ g(G') by Lemma <ref>.  As a side remark, consider Equation (15) of <cit.>. By taking degrees of the divisors we recover Lemma <ref>. Thus, one of the motivations for the RH-condition is a necessary and sufficient condition to have the ramification divisor be effective, as in the classical setting.
	
	

 §.§ Tropical morphisms
  By choosing a length function for the target G' we obtain a tropical morphism of metric graphs, in the sense of <cit.>.
	
	[tropical morphism]   Let : G → G' be a and ℓ_G' in C_G'. The corresponding tropical morphism is the unique map  from the source metric graph = (G,    e ↦ℓ_G' (φ(e)) /  m_φ(e)   ) to the target ' = (G', ℓ_G'), whose restriction to _e is a bijective linear function to '_(e), and that agrees with  on vertex points.
	
	
	Call  and (, ℓ_G') a model and a realization of , respectively. From the connectedness of G and G' we get that  is continuous. The length function for the source is constructed so the linear map |__e has integral slope m_(e). For a point x in the interior of an edge the slope is the multiplicity, and if x is a vertex then the multiplicity is computed with the balancing condition. With this, the count with multiplicities of points in the fibre (x') is independent of  x' in ', and moreover agrees with the degree of any model of . 
	
	Definition <ref> differs from <cit.> because here our focus is on combinatorial constructions. In the previous paragraph we have argued that the map = (, ℓ_G') is a tropical morphism in the sense of <cit.>. Conversely, given a tropical morphism :→', we choose vertex sets 𝒢, 𝒯 on , ' such that  induces a _ess. 
	
	[essential model] 
		Let :→' be a tropical morphism, and ℰ, ℰ' the essential vertices of , ' respectively. Set 𝒯 = ℰ' ∪(ℰ) and 𝒢 =   (𝒯). By Lemma <ref> these are vertex sets, so let (G, ℓ), (G', ℓ') be the induced realizations of , '. Define a map 
		_ess	 : G → G' by sending a vertex of  to the corresponding vertex in ', 
		and for each edge e in E(G) choose a point x in _e and map e ↦ e', where e' is such that (x) is in '_e'. As index map take m__ess = ℓ' ∘(e)/ℓ(e).
	
	
	 
		The pair (_ess, ℓ') realizes , and is minimal in the sense that for any other model of , the induced vertex sets on , ' arise as edge subdivisions of 𝒢, 𝒯.
	
	
	
		We claim that _ess is well defined and is a . Let x in _e and e' in E(G') such that (x) is in '_e'. We show that e' only depends on e. To see this, take x_1, x_2 in _e. There is a topological path P (the image of a continuous map ω: [0,1] →) from x_1 to x_2 contained in the interior of _e, so in particular it does not pass through any vertex of 𝒢. Thus, (P) does not pass through any vertex x' of 𝒯 (otherwise P would pass through some vertex in (x') ⊂𝒢). Hence, (x_1) and (x_2) are in the same '_e'. The index map m__ess records the slope of |__e, thus the balancing condition, non-degeneracy, and the RH-condition are satisfied. It is then clear that (_ess, ℓ') realizes .
		
		Regarding minimality, as are graph morphisms we get that (V(G)) ⊂ V(G'). Thus, in every model  of  the vertex set of ' has to contain (ℰ) (and ℰ'). Furthermore, the non-degeneracy condition of  implies that the fibre of a vertex of G' is a subset of V(G). So a vertex set of  has to contain (𝒯). 
	
	
	
	
	
	
	
	
	
	

 §.§ Isomorphisms, tropical modification, combinatorial type
   Recall that in the space g we consider metric graphs in an equivalence class under isometry and tropical modification. This setup extends naturally to tropical morphisms. 
	
	
	[isomorphism of graph morphisms]  
		
		Let _1: G_1 → G'_1 and _2: G_2 → G'_2 be graph morphisms. An isomorphism from _1 to _2 is given by graph isomorphisms (γ, γ') for which Diagram <ref>.a commutes. If additionally _1 and _2 are , then (γ,γ') is an isomorphism of if the index map m__2 pulls back to m__1, namely Diagram <ref>.b commutes. Let _1 and _2 be tropical morphisms. An isomorphism from _1 to _2 is given by isometries Ψ and Ψ' for which Diagram <ref>.c commutes.
		
		
		[b]0.9
			0.3
				
    G_1 [r,"_1"] [d,swap,"γ"]    
    				G'_1 [d,"γ'"] 
    
    				G_2 [r,"_2"]     G_2'

				
				(a)
			
			0.3
				
    G_1 [d,swap,"γ"] drm__1
    
    				G_2 [r,swap,"m__2"]    ℤ_≥ 0

				
				(b)
			
			0.3
				
    _1 [r,"_1"] [d,swap,"Ψ"]    '_1 [d,"Ψ'"] 
    
    				_2 [r,"_2"]    _2'

				
				(c)
			
			
			
			diagram
			thm
			
			Diagram <ref>
		
	
	
	
	We now define equivalence under tropical modification for a  : G → G'. We say that the fibre (x'), for x' in G', is dangling if all its elements are dangling in G. Let D ⊂ E(G') be the set of edges with dangling fibres. Consider the subgraph Ĝ'  of G' induced by E(G') ∖ D. The preimage (Ĝ') is not necessarily a connected graph (see Example <ref>), but the definition of dangling implies that there is only one connected component with non-zero genus. Call it Ĝ. Restricting  to Ĝ gives a : Ĝ→Ĝ' that we call the deletion of dangling fibres of . See the Section Modifications in <cit.> for some examples. Tropical modification identifies  with .
	
	The same definitions and constructions can be carried out on the metric side for a tropical morphism = (, ℓ). A fibre (x) is dangling if all its points are dangling, and the deletion of dangling fibres  is the restriction of  to the connected component with non-zero genus which remains after deleting all points belonging to dangling fibres. One can check that (, ℓ|_Ĝ') is a realization of . Tropical modification identifies  with . It is straightforward to see that if _1 and _2 are equivalent under tropical modification, then _1 and _2 are isomorphic, thus  is a canonical representative for the equivalence class of  under isomorphism and tropical modification. Thus, it is natural to use a model of  as the combinatorial type of the class of :
	
	 
		Let  be a tropical morphism and  its deletion of dangling fibres. The combinatorial type of  is the model _ess of  constructed using Construction <ref>.
	
	
	Alternatively, one can use Construction <ref> on  to get (_ess, ℓ_ess) and delete all dangling fibres to get _ess. It is straightforward to see that _ess is isomorphic to _ess. So (_ess, ℓ_ess|_Ĝ_ess') realizes . We call the values of ℓ_ess|_Ĝ_ess' the parameters of . Thus, we say that  moves in the cone C_ with dimension C_ and depends on |E(Ĝ_ess')| many parameters. 
	
	
	
	
	
	

 §.§ Cone of sources
 
	We associate to a  the set C_ of sources (modulo tropical modification) obtained by varying the parameters. By Section <ref> this is an invariant under isomorphism and tropical modification.  We say that  realizes  if ∈ C_ (recall that  is obtained by deletion of dangling points). We introduce an important linear map to see that C_ has the structure of a polyhedral cone. Another important invariant under tropical modification of   is the combinatorial type of its source, which we denote by H().
	
	
	Let : G → G' be a , H = H(), and g = g(H). 
	Consider a length function ℓ_G' in C_G'; the tropical morphism = (, ℓ_G'); and = (H, ℓ_H) where (H, ℓ_H) is the deletion of dangling elements of the source of . Extend the map sending ℓ_G' to ℓ_H (defined from ^E(G') to ^E(H)) to a linear map A_ from 
	^E(G') to ^E(H). We call A_ the edge-length map. Recall from Section <ref> that by choosing labellings λ_H and λ_G' we identify ^E(H) and ^E(G') with ^|E(H)| and ^|E(G')|. This gives an |E(H)| × |E(G')|-matrix for A_.
	
	We define the cone of sources C_ to be A_(^E(G')). It
	parametrizes, up to tropical modification, the sources of tropical morphisms with model . Let  be a tropical morphism with model . We say that  moves in the cone C_. 
	
	Note that C_ =  A_≤ 3g - 3. We are interested in tropical morphisms  that move in a cone whose dimension equals the number of parameters defining . Namely, those for which A_ = |E(G')|. We say that these cones, and , have full-rank.
	If  has full-rank then there is a unique preimage A_(ℓ_H) for ℓ_H. In other words, a unique tropical morphism (,  A_(ℓ_H)) whose source retracts to = (H, ℓ_H). 
	
	
	
	

 §.§ Combinatorial types
  Let : G → G' be a , and  a tropical morphism with model . Recall that the combinatorial type of  is the essential model _ess of the deletion of dangling points . Now we give an analogue of Lemma <ref> for tropical morphisms; namely we give combinatorial conditions that characterize the that show up as combinatorial types. We also show that with full-rank satisfy these conditions.
	
	 
		A combinatorial type of  is a  : G → G' without dangling fibres and such that ∑_A ∈(v) r_(A) ≥ 1 for every divalent v in V(G').
	
	
	  Figure <ref> illustrates two _A, _B. They are equivalent under tropical modification, and _B is a combinatorial type of . Note that the degree is not invariant. 
		
		
		

		
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			[t].45 
				
				_A			
			
			[t].45 
				
				_B
			
			
			figure 
		
	
	
	
	
	 
		Let  be a tropical morphism, and : →' be a tropical modification of , such that  has a model : G̅→G̅' that is a combinatorial type of . Then  is isomorphic to  and  is isomorphic to _ess.  
	
	
	
		By definition  has no dangling fibres. So by Lemma <ref> we only need to show that the points of ' corresponding to vertices of G̅' are in the set 𝒯, with 𝒯 as in Construction <ref> (the union of the set of essential vertices of ' with the image of the set of essential vertices of ). This means proving that for every divalent v in V(G̅') there is a vertex A in (v) such that A ≥ 3. 
		Note that for A in (v) we have r_(A) =  A - 2 - m_(A)( v - 2) =  A - 2. By the balancing condition there are no monovalent vertices in (v).
		Thus, ∑_A ∈(v) r_(A) = ∑_A ∈(v) A - 2 ≥ 1 if and only if there is at least one vertex A in (v) with A ≥ 3.
	
	
	
	
	Combinatorial types of are canonical representatives in the equivalence class under tropical modification: the discussion after Definition <ref> tells us how to construct them, and Lemma <ref> ensures their uniqueness.
	
	We call the sum ∑_A ∈(v) r_(A) the change at v, and denote it by v. We have seen in the proof of Lemma <ref> that divalent v with v = 0 arise as edge-subdivisions of the essential model. Thus, for divalent v if t, t' are the edges incident to v, then morally the fibres of t, t' are equal (each A in (v) has valency 2, and the two edges have the same index m_). This implies the following result, which is our motivation for calling this quantity “change”.
	
	 
		Let v in V(G') be divalent, and A = {t,t'}. If v = 0, then the columns of A_ corresponding to t, t' are equal.
	
	
	
		If : G → G' has full-rank, then  is a combinatorial type of .
	
	
	
		Let H be the combinatorial type of G, and A_: C_G'→ C_H be the edge-length map. Suppose t in E(G') has a dangling fibre. Then varying ℓ_G'(t) does not affect A_(ℓ_G'), contradicting that A_ has full-rank. So  has no dangling fibres. Moreover, by Lemma <ref> we have that  has no divalent vertices with v = 0.
	
	
	Finally, we prove a combinatorial property that puts the condition v ≥ 1, for divalent vertices, in a wider context.
	
	 
		If  is a combinatorial type, then v +  v ≥ 3 for all v in V(G').
	
	
	
		We only need to show that v ≥ 2 for monovalent vertices v of G'. Let v be monovalent. Note that a monovalent vertex of G is dangling. Since G does not have dangling fibres, pick A in (v) with A ≥ 2. For such A we have that r_(A) =  A - 2 - m_(A) ( v - 2) =  A - 2 + m_(A). Since v is monovalent we have that m_(A) ≥ A, thus r_(A) ≥ 2 and by the RH-condition this implies that v ≥ 2.
	
	
	
	
	
	

 §.§ Dimension Formula
   Let : G → G' be a . We prove a formula relating the number |E(G')|, the degree d =, and the genera of the source and the target of . Note that |E(G')| is an upper bound for C_. Hence, we call this result the dimension formula.
	
	[total change]  
		Let : G → G' be a . Then
		
    ∑_v ∈ V(G') v = 2g(G) - 2g(G') · d + 2d - 2.
 
	
	
	
		This is a consequence of Lemma <ref> (Riemann-Hurwitz formula).
	
	
	
	  
		Let : G → G' be a . Then
		
    |E(G')| + ∑_v∈ V(G') ( v +  v - 3) = 2g(G) - g(G')·(2d-3) + 2d - 5.
 
	
	
	
		The ingredients are the fact that the sum of valencies gives twice the number of edges, and Lemma <ref> (total change). We compute:
		
    |E(G')| + ∑_v∈ V(G') ( v +  v - 3)    = 3|E(G')| - 3|V(G')| + ∑_v∈ V(G') v
       = 2g(G) - g(G') · (2d-3) + 2d - 5.

		
	
	
	We regard the second term on the left hand side in Proposition <ref> as a correction term. Lemma <ref> gives us a choice in the equivalence class under tropical modification of  for which the correction term is non-negative. With this we get an upper bound:
	
	 
		Let : G → G' be a . Then 
    C_≤ 2g(G) - g(G')·(2d-3) + 2d - 5.

	
	
	
		Let :Ĝ→Ĝ' be a combinatorial type of that is equivalent under tropical modification to . By Lemma <ref> and the dimension formula we get
		
    C_≤ |E(Ĝ')| ≤ 2g(Ĝ) - g(Ĝ')·(2d-3) + 2d - 5.
   
		By the properties of tropical modification we have that C_ =  C_, g(Ĝ) = g(G), and g(Ĝ') = g(G'), proving the desired bound. 
	
	
	
	

 §.§ Tree gonality
  To close this section we use the bound given by the dimension formula to reprove a result on tree gonality that appeared in <cit.>. 
	We go on to outline what is the central problem for the rest of this article, and our strategy to tackle it.
	
	
		The tree gonality of a metric graph  is the minimum degree of a tropical morphism from some tropical modification of  to some metric tree. 
	
	
	Let : G → T be a to a tree. Every metric graph  in C_ has tree gonality at most . From now on we only work with : G → T to trees. For such maps Corollary <ref> gives that C_≤ 2g + 2d - 5. Thus:
	
	 
		Let g, d be positive integers. The locus of metric graphs in g with tree gonality at most d has dimension at most min(2g + 2d - 5, 3g-3). 
	
	
	
		Let  be in g,  a tropical modification of , and : → a tropical morphism to a tree with ≤ d. Then ∈ C_, where : G → T is the combinatorial type of . By the preceding discussion and since g = 3g-3 we get C_≤min(2g + 2d - 5, 3g-3). Note also that |E(T)| ≤min(2g + 2d - 5, 3g-3), and that |E(G)| ≤· |E(T)| ≤ d · |E(T)|, so there are finitely many possibilities for G, T, and consequently also for . This gives the result.
		
	
	
	Since 3g-3 ≤ 2g + 2d - 5 implies that ⌈ g/2+1 ⌉≤ d, we get that most metric graphs in g have tree gonality at least ⌈ g/2+1 ⌉. This is one of the two inequalities that comprises Theorem 1 in <cit.>. The other inequality, namely that metric graphs have gonality at most ⌈ g/2+1 ⌉, was shown constructively for certain families of graphs, one family for each combinatorial type:
	
	 
		Let H be a trivalent combinatorial type and g = g(H). Then there exists a  such that =  ⌈ g/2+1 ⌉, H() = H and C_ = 3g - 3.
			
	
	Theorem <ref> is proven constructively in Section 4 of <cit.>. When g is even, these families have the following property: 
	
	  A , with  g = g(H()), is full dimensional if 
    C_ = 3g - 3 = 2g + 2d - 5.

		
	
	
	The pairwise equalities in the definition of full-dimensional have the following consequences:
	
		
  * C_ = 3g-3 implies that H() is trivalent and the tropical morphism moves in a space of the right dimension, a necessary condition to hope to realize all sources in C_H().
		
  * C_ = 2g + 2d - 5, C_ = 3g-3, and the dimension formula together imply that |E(T)| = 3g-3, that  has full-rank, and v +  v = 3 for all v in V(T).
		
  * 3g - 3 = 2g+2d - 5 implies that g is even and d = g/2 + 1.
	
	
	These observations give a practical criteria to check whether  is full-dimensional:
	
	 
		A  is full-dimensional if and only if A_ is a nonsingular (3g-3) × (3g-3) matrix.
	
	
	
		Lemma <ref> and the calculation done in the fifth case of Example <ref> shows that the map _A in Example <ref> is full-dimensional.
	
	
	Let H be a trivalent combinatorial type with even genus, and  the full-dimensional given by Theorem <ref>. Our aim here, in Part I of this series of two articles is to introduce a deformation procedure that, when applied to , yields full-dimensional 1, 2, and so on, with cones C 1, C 2, and so on, which together cover C_H, showing that all metric graphs with combinatorial type H have gonality at most g/2 +1. For the remaining we direct our efforts to the construction of this deformation procedure. We are not limited by the condition that g is even, since the odd genus case follows from the even genus case, see Subsection <ref>.
	
	
	
	As a sneak peek, we mention that in Part II of this series of two articles the main problem is the study of the set 𝔽𝔻_g of full-dimensional whose source has genus g. A full-dimensional has full-rank, so it is a combinatorial type. By the argument given at the end of the proof of Corollary <ref> we have that 𝔽𝔻_g is finite. 
	
	We glue together the cones corresponding to elements in 𝔽𝔻_g in a construction akin to that of g and obtain a space d g. There is a projection Π from d g to g where a tropical morphism  is mapped to its source. We show that Π is a surjection,  that d g is connected in codimension-1, and that the fibres of Π have Catalan-many points when counted with certain multiplicity. This opens the possibility of proving the upper bound on gonality without using Theorem <ref>: instead, we only need to study the fibre Π() for one particular kind of metric graph , and the rest is deformation.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

§ GLUING DATUMS
  
	Let : G → T be a to a tree. That the fibre (x) has  elements counted with multiplicity, and that  is a graph homomorphism, leads us to regard G as the graph resulting from taking  copies of T, and identifying together certain vertices and edges between copies. In this view the index map m_ records how many copies of T were glued together in a particular place. We make this notion precise with the combinatorial gadget of a gluing datum. 
	
	
	
	
	
	

 §.§ The gluing datum
   gluing datums are a tool to ease the visualization of ; to do book-keeping on edges of G in the process of deformation of  (see Sections <ref> and <ref>); and to write computer programs (see <cit.> for code). 
	
	
	
	
	[gluing datum] Let T be a finite tree, d be a positive
		integer, and ∼ be an equivalence relation on W = T × [d], where [d] = {1,…,d}. Write x k for the classes of W/∼, where x ∈ V(T) ∪ E(T). The triple (T, d, ∼) is a  gluing datum when ∼ satisfies these properties:
		
			
  * Verticality: If (x, i) ∼ (x', i'), then x = x'. Each x in T defines a relation ∼_x on [d] with i ∼_x j if (x, i) ∼ (x, j). We say that ∼_x is the gluing relation above x. By verticality these relations determine ∼.
			
			
  * Refinement: If v is in V(T), and t is incident to v, then ∼_t is a refinement of ∼_v.
			
  * Connectedness: For any two classes xk and x'k' there is a sequence x_0i_0 = xk, x_1i_1, …, x_ri_r = x'k' such that for each q the elements x_q, x_q+1 are incident (so one of them is a vertex and the other an edge), and there is at least one j such that (x_q, j) ∈x_qi_q and (x_q+1, j) ∈x_q+1i_q+1.  
			
  * Riemann-Hurwitz condition: For v in V(T) and i in [d] let A ⊂ [d] be the class of i under the relation ∼_v. Let l =  v and v = {t_1, …, t_l}.
			
			By the refinement property the relation ∼_t_q above t_q partitions A into k_q sets.
			Then
			
    (k_1 + k_2 + … + k_l) - 2 ≥ |A| · (l-2).

		
	
	
	
	
	
	Let M = (T, d, ∼) be a gluing datum. Write | x k| for the cardinality of the equivalence class x k. We abuse notation and say i in x k to mean (x,i) in x k. By verticality this causes no confusion. A consequence is to regard x k as a subset of [d], for the purpose of comparing classes. We do, but to avoid confusion it is pointed out in every such instance of this use that x k is momentarily being regarded as a subset of [d], instead of a class of W/∼.   
	
	We call T the base tree of M. We visualize gluing datums by drawing d copies of T on top of each other, and imagining all of them above T. Curved lines and shaded regions indicate where vertices or edges are identified. A class x k is said to be above x. By the verticality condition, one can only glue elements of W that are above the same element of the base tree, see Figure <ref>. 
	
	
	The quotient G = W/∼ has the following natural structure of a loopless graph: the classes x k in W/∼ with x in V(T) are the vertices of G; those with x in E(T) are the edges of G; an edge t i in E(G) is incident to a vertex v k in V(G) if and only if t and v are incident in T and i ∈ v k. This is a well defined graph, by the verticality and the refinement properties. It is connected, by the connectedness property. It has no loops by the verticality property and the fact that T is a tree. Consider the natural map φ_M: G → T given by φ_M  x k = x, and the index map m__M x k = | x k|. 
	
	
	[Prop. 5, <cit.>] For a gluing datum M = (T, d, ∼), the map φ_M  x k = x with index map m__M x k = | x k| is a of degree d.
	 
	
		
		We prove that  is a by showing the balancing condition; the other requirements are immediate.
		Let A be a vertex of G. Set v = (A). Fix t in v. Recall that by the refinement property the relation ∼_t above t partitions A. Thus, if e_1, …, e_r are the classes in E(A) above t, then |A| = |e_1| + … + |e_r|. Since m_(e) = |e| we get that
		
    m_φ(A) = |A| =  ∑_e ∈ A 
    φ(e) = t m_φ(e).
 
		
	
	We illustrate the previous concepts in the following example. 
	
	
	  
		Let M = (T,3,∼) be a gluing datum with T and ∼ as in Figure <ref>.
		
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    ∼_v_1 = ∼_v_2 = {{1,2},{3}}
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       else ∼_x = {{1},{2},{3}}.

			 
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			[t].32
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			[t].32 
				
 
					 A visualization of M

			
			
			figure 
		
		
		

		
		In the visualization we have rendered dangling edges with dotted lines and lighter colour. The copies of T are numbered from top to bottom. We also have labelled with A, B, …, F the vertices of G that have non-dangling valency equal to 3. The other non-dangling vertices have non-dangling valency equal to 2. Note that the map _M is isomorphic as a to the map _A of Example <ref> (see also Example <ref>).
	
	
	
	We denote by A the set of non-dangling edges of G incident to a vertex A and define the non-dangling valency A as |A|. 
	Observe that A = 0 if and only if A is dangling, otherwise A ≥ 2. In the visualization of M in Example <ref> the darker coloured edges make up the following graph:
	
	
		Let H(M) be the following graph: V(H(M)) is the set of vertices A of G such that A ≥ 3; and E(H(M)) is the set of paths of G whose ends are in V(H(M)) and interior vertices have non-dangling valency equal to 2.
	
	
	Recall that H(_M) is the combinatorial type of the source of the _M associated to M. It is straightforward to see that H(M) and H(_M) are isomorphic. Observe also that the edge e of G is either dangling, or there is a unique h(e) in E(H(M)) such that e ∈ h(e). We say that h in E(H(M)) passes through a class x k of G if x k ∈ h and x k is not and end of h. If h passes through x k, then we say that h passes above x. This gives a natural way to also visualize H(M) above T, as in Example <ref>.
	
			
		Figure <ref> illustrates G and H(M) for the gluing datum M of Example <ref>. We kept the labelling for vertices with non-dangling valency greater than 2 from the visualization of M in Example <ref>, for ease of comparison. 
		
		

		
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			[t].6 
				
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				H(M)
			
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		Unlike our approach here where a gluing datum gives rise to a , the definition of gluing datum in <cit.> gives rise to a tropical morphism : →. To go from a gluing datum as in <cit.> to one as in our setting we need to remove the metric information by taking a model T of  where for every edge t of T the relation above each interior point x in _t equals some fixed relation ∼_t. In the terminology of the Definition 4 of <cit.> a possibility for such a model is the set of points from the essential model of  together with the image under  of the set of monovalent points in the metric forest  
    {u ∈  |  ∃ x ∈ with ψ_i(x) ∼ψ_j (x)   for distinct  i, j ; and  u = ψ_i(x)/∼},
 where the map ψ_i sents  to its i-th copy.
	
	
	
	    Given a degree-d :G → T we construct a gluing datum M with base tree T such that  is isomorphic to _M. We use four steps:
		
			
  * Choose a leaf v of T and a map χ_v : (v) →𝒫([d]), where 𝒫([d]) is the power set of {1,…,d}, such that |χ_v(A)| = m_φ(A) for A ∈(v) and (χ_v) is a partition of [d].
			
  * Choose incident x, x' in T such that a map χ_x has been chosen for x and no map has been chosen for x' yet.
			
  * Choose a map χ_x' : (x') →𝒫([d]) such that |χ_x'(X')| = m_φ(X') for X' ∈(x'), (χ_x') is a partition of [d]; and for every pair of incident elements (X',X) in (x') ×(x) we have that χ_x'(X') ⊂χ_x(X) if x' is an edge, or χ_x'(X') ⊃χ_x(X) otherwise.
			
  * Repeat step 2 and 3 until every x in T has a χ_x.
		
		Take M = (T,d,∼), where ∼_x is defined by the partition (χ_x).
		
	
	
	In Step (3) of Construction <ref> the choice is possible since  satisfies the balancing condition. It is straightforward to check that the gluing datum (T,d,∼) has the properties we desire. This construction is essentially the one given in  Proposition 6 of <cit.>. 
	
	
	
	
	
	

 §.§ Isomorphism classes of and gluing datums
  
	The choices that are crucial in Construction <ref> happen at Steps (1) and (3). We identify all the possibilities using a notion of isomorphism of gluing datums. It turns out that with the right notion of isomorphism there is a bijection between isomorphism classes of and isomorphism classes of gluing datums. This completes the setup to study via gluing datums. 
	
	Observe that applying a fixed permutation π of [d] on all relations ∼_x gives a different gluing datum, but does not change the isomorphism type of G, nor the index map for the edges of G, nor which edges of G lie above a particular edge of T. We call this operation a tree-swap. It enables us to relate two distinct choices done at Step (1) of Construction <ref>.
	
	Choose a vertex v of T, and let S be the vertex set of one of the connected components of T ∖{v}. Let T_br be the graph induced by the vertex set S ∪{v}. We call it a branch of T. Choose i and j such that i ∼_v j and swap i with j in all relations above T_br. We call this a branch-swap. A series of branch-swaps enables us to relate two distinct choices done at Step (3) of Construction <ref>.
	This motivates the notion of isomorphism of gluing datums. In the following definition we write x i_M for a class of M, to avoid confusions, and in general we regard x i_M as a subset of [d].
	
	
	[gluing datum isomorphism]  An isomorphism from M = (T, d, ∼) to M' = (T', d, ∼') consists of a graph isomorphism τ:T → T', and for each x in T a permutation π_x of [d] satisfying two properties:
		
			
  * class-preserving: π_x (   x i_M ) and τ(x)π_x(i)_M' are equal for all x in T, i in [d].
			
  * incidence-preserving: for all v in V(T), t in E(v), and i in [d] we have that t j_M ⊂ v i_M if and only if τ(t)π_t(j)_M'⊂τ(v)π_v(i)_M'.  
			
	
	
	
	  
		Figure <ref> presents three isomorphic gluing datums. On the left, M from Example <ref>. Next, in M' we swapped the middle and the bottom tree. Finally, in M” we also swapped the middle and the bottom tree, but only above the branch that is right to the vertex C. They all give rise to isomorphic .
		
		

		
		
			.3
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			.3 
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			.3 
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					(15,33) 0.8A
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			[t].3
				M			
			
			[t].3 
				M'			
			
			[t].3 
				M”
			
			
			figure 
		
		
		
	
	
	  The  isomorphism classes of gluing datums are in bijection with the isomorphism classes of from a graph to a tree.
	
	
	
		Let M, M' be gluing datums giving rise to graphs G, G' respectively, and = _M, ' = _M'. First suppose that τ, {π_x}_x ∈ T is an isomorphism from M to M'. Consider the natural map γ: G → G' given by γ (  x k_M ) = τ(x)π_x(k)_M'. By the class-preserving property we have that γ is bijective. By the incidence-preserving property we have that both γ, γ are graph homomorphisms. We claim that (γ, τ) is an isomorphism from  to '. Diagram <ref>.a commutes because
		
    ' ∘γ(  x k_M )    = ' (τ(x)π_x(k)_M')  = τ(x),
    τ∘ x k    = τ(x).

		Diagram <ref>.b commutes because by class-preserving we get that
		
    m_(  x k_M )    = | x k_M | = |τ(x)π_x(k)_M'|, 
    
    		m_'∘γ(  x k_M )    = m_'( τ(x)π_x(k)_M')= |τ(x)π_x(k)_M'|.

		
		
		Now suppose that γ, τ is an isomorphism from  to '. Since m_ = m_'∘γ, we get that | x k_M | = |γ (  x k_M )|. Thus, for each x in T we can choose a permutation π_x of [d] such that π_x (  x k_M ) = γ (  x k_M ). We claim that τ, {π_x}_x ∈ T is an isomorphism from M to M'. We have the class-preserving property by construction. Since γ, γ are graph homomorphisms, we have that t j_M, v i_M are incident if and only if γ (   t j_M  ), γ (   v i_M  ) are incident. Since ' ∘γ = τ∘, 
		we get that γ (  x i ) = τ(x)π_x(i). The previous two facts and the construction of G and G' together imply that t j_M ⊂ v i_M 
		if and only if τ(t)π_t(j)_M'⊂τ(v)π_v(i)_M'.
		
		Thus, we have shown that the map that sends the isomorphism class of M to the isomorphism class of _M is well defined and injective. Construction <ref> shows that the map is also surjective.
	 
	
	Now that the correspondence between and gluing datums is set up, we use the following notation for convenience:
	
	
		
		
  * C_M = C__M, the cone of sources.
		
  * A_M = A__M, the edge-length map.
		
  * (M, ℓ_T) = (_M, ℓ_T), a tropical morphism. 
		
  * We say that M has full-rank, or is full-dimensional, or is a combinatorial type if _M has full-rank, or is full-dimensional, or is a combinatorial type, respectively.  
	
	
	We also define the genus of a gluing datum to be g(H(M)). Likewise, the genus of a is the genus of its source.
	
	
	
	
	

 §.§ Local properties
  
	Let M = (T, d, ∼) be a genus-g gluing datum and  be _M : G → T. Ideally, a deformation procedure would modify a local part of M, namely the gluing relations above a particular edge of T and its endpoints. The resulting M' should be full-dimensional if we began with a full-dimensional M. The obstruction to this idea is that a local change in M leads to a change in A_M that may make the rank drop. Controlling the rank of A_M is a global condition, which is harder to check than local conditions.
	
	We defer the hard problem of constructing full-dimensional gluing datums to Section <ref>, and explore a different question here: what conditions are necessary for a gluing datum to be full-dimensional. It turns out that full-dimensional gluing datums have some remarkable combinatorial properties, which are easier to check than the rank. This allows us to easily discard many potential constructions as not full-dimensional. We present these combinatorial properties in Definition <ref>, after giving some auxiliary definitions.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	  Let M = (T, d, ∼) be a gluing datum.
		
			
  * A vertex v of T is change-minimal if v +  v = 3. 
			
  * A vertex A of G satisfies the no-return condition if there are at least two non-dangling edges in E(A) above different edges of T; in other words |(A)| is at least 2.
			
  * Let h = ⟨ A_0, e_1, …, e_μ, A_μ⟩ be an edge of H(M) (recall that this means that h is a path in G, both ends of h have  value at least 3, and the inner vertices of h have  value equal to 2). We say that
			h satisfies the pass-once condition if  restricted to the set {e_i ∈ h  | (e_i)  is not incident to a leaf} is injective.
		
	 
	
	
	  Let M = (T, d, ∼) be a gluing datum.
		
			
  * M is change-minimal if all vertices of T are change-minimal.
			
  * M satisfies the dangling-no-glue condition if | x k| = 1 for all dangling | x k|.
			
  * M satisfies the no-return condition if all non-dangling vertices A of G such that (A) is not monovalent satisfy the no-return condition. 				
			
  * M satisfies the pass-once condition if all edges of H(M) satisfy the pass-once condition.			
		
	 
	
	
	We proceed to describe informally the intuition behind each property and give an example before beginning with proofs. Being change-minimal is equivalent to saying that the correction term in the dimension formula equals zero, namely that |E(T)| = 2g + 2d - 5. By the discussion after Definition <ref> we have that full-dimensional implies change-minimal. Dangling-no-glue says that no identification is used in parts of G that do not affect ℓ_H(M). This is a sort of “efficiency” condition. No-return forbids edges of H(M) to change copies of T in any place except above a leaf of T. Pass-once eases the calculation of A_ (see Proposition <ref>).
	
	  The gluing datum of Example <ref> is full-dimensional (see the calculation at 5.a of Example <ref>). It can be checked that this gluing datum is change-minimal, satisfies dangling-no-glue, no-return, and pass-once; this agrees with our claim. Now we point out instances in the examples that do not fullfill some of the items in Definition <ref>.
		
		
		
			
  * Consider the map _B from Example <ref>. The second vertex from left to right in the target is the only change-minimal vertex, and above it the no-return condition is violated. The pass-once condition is violated by the edge passing above the middle edge of the target. The dangling-no-glue condition is violated by all of the dangling elements.
			
  * Consider the map _A from  Example <ref>. It does not satisfy dangling-no-glue because of the leftmost vertex in the target, and is not change-minimal. It satisfies no-return and pass-once.
		
	
	
	Instead of showing that full-dimensional implies dangling-no-glue, no-return and pass-once, we derive these conditions from being change-minimal and having full-rank. This opens the possibility of using these results in future work for studying those special graphs that have gonality less than ⌈ g/2 ⌉ + 1. We begin with a generalization of Lemma <ref>.
	
	[zero change]   Let v be in V(T), h an edge of H(M) such that (h) ∩ E(v) contains two edges t_1 and t_2, and  A_1,  …, A_r the vertices in h ∩(v). If the following conditions are true:
		
			
  * r_φ(A_q) = 0 for all 1 ≤ q ≤ r,
			
  * the ends of h are not above v,
			
  * |e| = 1 for dangling e in E(A_q) for all 1 ≤ q ≤ r,
			
		then in the edge-length matrix we have a_ht_1 = a_ht_2. 
	 
	
	Before proceeding to the proof, note that 
	the coefficient a_ht, for h in E(H(M)) and t in E(T), can be calculated as follows:     
	
    ℓ_H(M)(h) = ∑_e ∈ E(G) :   e ∈ h ℓ_G(e) = ∑_t ∈ E(T) (∑_e ∈ h:  (e) = t 1 |e| ) ℓ_T(t) = ∑_t ∈ E(T) a_htℓ_T(t) ,
  
	where the sum in the parenthesis equals zero if the index set is empty.
	
	
	 
		Since A_q is not an end of h we have that A_q = 2. Let A_q = vk_q, and e,e' be the two edges in A_q. Lemma <ref> and conditions (a) and (c) imply that 2|A_q| = |e| + |e'|, and since |A_q| ≥ |e|,|e| we get that |A_q| = |e| = |e'|, which makes the classes t_1k_q, A_q, t_2k_q equal as subsets of [d].  Thus, a_ht_1 = 1/t_1k_1 + … + 1/t_1k_r = 1/t_2k_1 + … + 1/t_2k_r = a_ht_2.
	
	
	Note that Condition (c) of Lemma <ref> is implied by dangling-no-glue. Recall that if v = 2, then r_(A_q) = 0 gives that A_q = 2. Hence, A_q is not an end of h, and there are no dangling edges incident to A_q, fulfilling Conditions (b) and (c). Now we prove dangling-no-glue.
	
	[change-minimal leaves] 			
		Let v in V(T) be a leaf that is change-minimal, A in (v) non-dangling, and t the edge incident to v. Since A is non-dangling A ≥ 2, and as v = 1 we get that |A| ≥ A. Thus, r_(A) =  A - 2 - |A|( v - 2) ≥ 2. Since v = 2 we get r_(A) = 2, A = 2, and |A| = 2. Thus, there is a unique non-dangling vertex above v, ∼_t is trivial, and  exactly one edge of H(M) passes above v and t. 
		
		This implies that in the edge-length matrix A_M the column corresponding to t has only one non-zero entry; it is a 2 in the row corresponding to the edge of H(M) passing above v. Thus, if M is change-minimal and has full-rank, then each edge of H(M) passes above at most one leaf; for otherwise A_M would have two equal columns, a contradiction.
	
	
	
	
	 
		Let M be change-minimal and full-rank, and A in V(G) dangling. Then r_φ(A) = 0.
	
	
	
	
		Let v = (A). By change-minimal r_(A) equals  0, 1, or 2. If r_(A) = 1, then Remark <ref> (change-minimal leaves) implies that v = 2, and so v =  1. Thus, if h in E(H(M)) passes through A' above w_0, then r_(A') = 0, because A' is non-dangling, so A'  A and r_(A') + r_(A) ≤ v = 1. Hence, Lemma <ref> (zero change) implies that the columns of A_M corresponding to t_1, t_2 incident to v are equal, contradicting that M has full-rank. If r_(A) = 2, Remark <ref> implies that A is non-dangling, a contradiction.
	
	
	  
		A change-minimal and full-rank M satisfies the dangling-no-glue condition.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
		Observation I: Let A be a vertex such that r_(A) = 0 and all elements of A have cardinality 1, except possibly one element e. Then by Lemma <ref> (formula for r_φ) we get that r_φ(A) = 2(|A| - 1) - (|e|-1). As r_φ(A) = 0 we get |e| + 1 = 2|A|. Hence, |A|=|e|=1 since |A| ≥ |e|.
		
		Now suppose that e_0 is dangling and |e_0| > 1. Choose an end A_0 of e_0. If A_0 is dangling then Lemma <ref> allows us to apply Observation I to choose e_1 in A_0, distinct from e_0, such that |e_1|>1. Repeat the previous step with e_1, and continue to construct a sequence of vertices A_0, A_1, and so on; since G is finite, either we arrive to a non-dangling A_r, or a vertex repeats in the sequence. Either way this means that after deleting e_0 the connected component of A_0 has a cycle. This argument can be applied to the other end of e_0 as well, giving a contradiction that e_0 is dangling. Thus, if e is dangling, then |e| = 1. If A is dangling, all edges in A are dangling, so they have cardinality 1, and the result follows from another application of Lemma <ref> and Observation I.
		
		
		
		
	
	
	Now we prove the no-return condition.
	
	
	[nd. r_ formula]  
		Let M satisfy dangling-no-glue, and A be in V(G). If e_1, e_2, …, e_r in E(G) are the non-dangling edges incident to A, then 
    r_φ (A) =  A - 2 + 2|A| - (|e_1| + … + |e_r|).

	
	
	
		This is a consequence of Lemma <ref> and the dangling-no-glue condition.
	
	
	
	[r1 implies no-return]  
		Let M satisfy dangling-no-glue, and A in V(G) be non-dangling with (A) ≥ 2. If r_φ(A) ≤ 1, then A satisfies no-return.  
	
	
	
		If |A| = 1 the result is clear. Assume that |A| ≥ 2. Let e_1, …, e_r be the edges in A. Note that r ≥ 2, since A is non-dangling. Assume that all e_q are above the same edge t, so |e_1| + … + |e_r| ≤ |A|. Lemma <ref> (nd. r_ formula) implies that r_φ (A) = r - 2 + 2|A| - (|e_1| + … + |e_r|) ≥ r - 2 + |A| ≥ 2, a contradiction.   
	
	
	
	  
		A change-minimal M that satisfies the dangling-no-glue condition also satisfies the no-return condition.
	
	
	
		Since M is change-minimal by Remark <ref> if r_(A) ≥ 2, then (A) is a leaf. 
		The corollary follows then from Lemma <ref>. 
	
	
	
	By dangling-no-glue, G looks like T locally around dangling vertices. We now give a local description for non-dangling vertices, assuming a change-minimal M. To make coming references to this result easier, we state all the possible cases, though several are straightforward corollaries of previous results. 
	
	  
		Let M be change-minimal and satisfying dangling-no-glue, A in V(G) with A either 2 or 3, v = φ(A), r =  A, and {e_1, …, e_r} =  A. Then exactly one of the following cases happens:  
		 
			
  * (r0-nd3): r_φ(A) = 0 and A = 3. Then  restricted to A is injective; v is 3; and |e_1| + |e_2| + |e_3| = 2|A| + 1.
			
			
  * (r0-nd2): r_φ(A) = 0 and A = 2. Then  restricted to A is injective; v is 2 or 3; and |e_1| = |A| = |e_2|. 
			
			
			
  * (r1-nd3): r_φ(A) = 1 and A = 3. Then there is a labelling of the e_i where (e_1) = (e_2) (e_3); v is 2; and |e_1| + |e_2| = |A| = |e_3|. 
			
			
			
  * (r1-nd2): r_φ(A) = 1 and A = 2. Then  restricted to A is injective; v is 2; and there is a labelling of the e_i where |e_1| = |A| and |e_2| = |A|-1. 
			
			
			
  * (r2-nd2): r_φ(A) = 2. Then (e_1) = (e_2); v is 1; |e_1| = |e_2|= 1; and |A| = 2.
		
	
	
	
		Change-minimal implies that r_φ(A) equals 0, 1, or 2, and that v equals 1, 2, 3 for v in V(T). By Remark <ref> (change-minimal leaves) r_φ(A) = 2 and A = 3 cannot be. So the described cases are all the possibilities. The same remark proves (r2-nd2). Note that M satisfies the no-return condition by Corollary <ref>.
		
		For the case (r0-nd3), no-return implies that either the three non-dangling edges are above different edges of T, or (up to labelling) e_2 and e_3 are above the same edge and e_1 is above another. In the latter case Lemma <ref> and |e_1| ≤ |A| imply 0 = 1 + 2|A| - |e_1| - |e_2| - |e_3| ≥ 1 + |A|  - |e_2| - |e_3|, but |e_2| + |e_3| ≤ |A|, a contradiction.  Hence e_1, e_2, e_3 are above distinct edges of T, so v = 3. Lemma <ref> implies the last formula.   
		
		The remaining follows from change-minimal, dangling-no-glue, Lemma <ref>, and the gluing datum axioms.
	
	
	
	 
		Let M be change-minimal and satisfy dangling-no-glue. Let P = ⟨ A_0, e_1, …, e_μ, A_μ⟩ be a path of G such that A_i is non-dangling and (A_i) ≥ 2 for each i. Then ⟨(A_0), (e_1), …, (e_μ), (A_μ) ⟩ is a path of T.
	
	
	
		Since  is a graph morphism, and by the local properties of Proposition <ref> , we get that consecutive elements of (P) := ⟨(A_0), (e_1), …, (e_μ), (A_μ) ⟩ are incident, and distinct. Since consecutive elements of (P) are distinct, if (P) repeats an element, then we get a cycle. As T is a tree we conclude that (P) is a path.
	
	
	Now we move on to the pass-once condition.
	
	
	  Let M be change-minimal and have full-rank, u and v be divalent and monovalent vertices of T, respectively, with u = {t', t}, v = {t}. Then there is exactly one edge of H above t and one above t'; they are a loop and a bridge, respectively.
	   
	
	
	
		Swap trees so that 1 ∼_v 2. Remark <ref> (change-minimal leaves) yields that e_1 =  t 1 and e_2 =  t 2 are two distinct edges of G, with |e_1| = |e_2| = 1, and are the only non-dangling edges above t. This implies, by the no-return condition, that u 1 and u 2 are the only non-dangling classes above u (it is possible that u 1 =  u 2). Let h be the edge of H containing e_1, e_2. If both u 1 and u 2 have non-dangling valency equal to 2, then the only edge of H passing above t, and t' is h. Thus, a_ht and a_ht' are the only non-zero entries in the columns of A_M corresponding to t and t', respectively, a contradiction. Therefore, assume without loss of generality that u 1 = 3. As u = 2 and |e_1| = 1, by Case (r1-nd3) of the local properties we must have that u 1 has two edges above t; namely e_1, e_2, that make the loop; and one edge above t', the bridge.
	  
	
	
	 A change-minimal and full-rank M satisfies the pass-once condition
	
	
	
		Let h = ⟨ A_0, e_1, …, e_μ, A_μ⟩ be an edge of H(M). If h does not pass above a leaf, then we are done by Corollary <ref>. Assume h passes above a leaf. By Remark <ref> (change-minimal leaves) there is a single vertex A_i ∈ h such that v := (A_i) is a leaf. Thus, h passes at most twice above any edge of T by Corollary <ref>. We also get that (e_i) = (e_i+1) from Remark <ref>. Let t_1 = (e_i). Suppose that h violates the pass-once condition. That is, for some j,k ∉{i,i+1} we have (e_j) = (e_k). We get that (e_i-1) = (e_i+2), because otherwise (⟨ A_0, … A_i ⟩) and (⟨ A_i, … A_μ⟩) would intersect only at t_1 by Corollary <ref> and the fact that T is a tree, which contradicts that (e_j) = (e_k). Let t_2 = (e_i-1).
		
		Label the copies of T such that 1 ∼_v 2. Label h with 1. Lemma <ref> implies that w = 3. Let w be the other end of t_1, and t_3  be the other edge incident to w. We investigate the classes above t_1, t_2 and t_3. Observe that t_3 1 and t_3 2 are dangling. Change-minimal and Remark <ref> imply that |t_1 1| = |t_1 2| = 1. All the vertices of G above w have r_φ-value zero. Thus, case (r0-nd2) of the local properties on w 1 and w 2 implies that |t_2 1| = |t_2 2| = 1.  Hence, a_1,1 = 2, a_1,2 = 2, a_1,3 = 0. As t_1 k is dangling for k ≥ 3, the other non-dangling vertices above w have  equal to 2. Lemma <ref> (zero change) implies that a_i,2 = a_i,3 for i ≥ 2.  See Figure <ref>, dangling edges shown dotted. 
		
		
		
		
			[b].5
				[scale=1.5]/283.pdf
					(35,-6) 1v
					(55,24) 1w    
					(47,9) 1t_1
					(25,24) 1t_2
					(77,24) 1t_3
					(-2,47) 1e		
				  
			
			[b]0.5
				
    a_1,1 = 2 
    				     a_1,2 = 2
    				     a_1,3 = 0
       a_i,1 = 0,          
    				a_i,2 = a_i,3,  
       for  i ≥ 2.

				 
			
			figure 
			
		
		

		
		Let 𝐚_1, 𝐚_2, and 𝐚_3 be the first three columns of A_M. Note that 𝐚_2 = 𝐚_1 + 𝐚_3, contradicting that M has full-rank.
	
	
	
	
	  
		Let M = (T,d,∼) satisfy that the leaves of T are change-minimal. Then M satisfies the pass-once condition if and only if for all h in E(H), and t in E(T) the entry a_ht is:
		
			
  * 2 if h passes above a leaf v in V(T), and t is incident to v.
			
  * 1/|t k| if h passes through some edge t k of G, t not incident to a leaf. 
			
  * 0 otherwise.
		
	
	
	
		Part (a) follows from Remark <ref>. Part (b) is true if and only if pass-once is. 
	
	
	
	
	
	
	
	
	
	
	
	

§ EXAMPLES
   
	
	[3 loops on a loop]   Let = (H, ℓ_H) be a loop of 3 loops (as in <cit.>), with H, vertex and edge labellings shown in Figure <ref>. Set y_i = ℓ_H(h_i), z_i = ℓ_T(t_i).
		
		

		
		
			[scale=0.6]07-cycles2.pdf  
				(78,53) 0.8y_1
				(47,3) 0.8y_2
				(15,53) 0.8y_3
				
				(72,24) 0.8y_4
				(89,11) 0.8y_5
				(23,24) 0.8y_6
				
				(5,10) 0.8y_7
				(47,66) 0.8y_8 
				(47,89) 0.8y_9
				
				(67,78) 0.8A
				(90,35) 0.8B
				(70,0) 0.8C
				(23,0) 0.8D
				(3,35) 0.8E
				(27,78) 0.8F
			
			
			

			H
			figure 
		
		
		

		
		Assume without loss of generality that y_3 = max(y_1, y_2, y_3), y_4 ≤ y_5, y_6 ≤ y_7, and y_8 ≤ y_9. We show 9 cases of glueing datums which realize . These depend on how large the length y_3 is, compared to several other linear combinations of lengths. 
		Each case depicts: the base tree T with an edge labelling; the glueing datum which realizes ; and its edge-length matrix using as labelling for H the one shown in Figure <ref>. The odd cases are full-dimensional, the even cases are not. In two consecutive odd cases the edge-length matrices differ only by one column. 
		We compute A_M to show these are indeed full-dimensional.
		
		
		
		
		  
			
  * First case: if y_3 < y_1 + y_2, then M_1 in Figure <ref> realizes .
			
			

			
			
				.32
					[scale=1]/046.pdf    
						(42,23) 0.8z_1 
						(65,46) 0.8z_2 
						(30,46) 0.8z_3 
						(41,9) 0.8z_4 
						(18,9) 0.8z_5 
						(82,32) 0.8z_6 
						(82,50) 0.8z_7
						(12,32) 0.8z_8 
						(12,50) 0.8z_9 
					 
					
				
				.32 
					/208.pdf    
						(15,58) 0.8A
						(25,24) 0.7B
						(25,11) 0.7C
						(80,45) 0.7D
						(80,71) 0.7E
						(15,71) 0.7F
					 
					
				
				.25 
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				[t].32
					T_1			
				
				[t].32 
					M_1			
				
				[t].25 
					A_1 =  128
				
				figure 
				

			
			
			
			
			
  * Second case: if y_3 = y_1 + y_2,  then M_2 in Figure <ref> realizes .
			Note that edge t_1 was contracted.
			
			
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				[t].32
					T_2			
				
				[t].32 
					M_2			
				
				[t].25 
					A_2 

				
				figure 
			
			
			

			
			
			
  * Third case: if y_1 + y_2 < y_3 < y_1 + y_2 + y_4, Figures <ref> and <ref> show realizations of .
			 
				
  * Case 3.a 
				
				
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					[t].32
						T_3 a			
					
					[t].32 
						M_3			
					
					[t].25 
						A_3 =  -64
					
					figure 
				
				
				
  * Case 3.b
				
				
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					[t].32
						T_3 b			
					
					[t].32 
						M_3			
					
					[t].25 
						A_3 

					
					figure 
				
				

				
			
			
			Here  is realized by two tropical morphisms (distinct ones in general, as we argue next), yet at the combinatorial level the gluing datum is the same. We drew M_3 twice to illustrate which of y_4, y_5 is longer; and to show the labelling of the base tree. The two length functions z a, z b are related by swapping z_4 and z_5. 
			
			We claim that a = (_3,  z a) and b = (_3,  z b) are distinct tropical morphisms if and only if z_4, z_5 are distinct lengths. Set v =  (B). 
			Since z_4, z_5 are distinct, one of the two edges between B, C is longer than the other, and there is a preimage of v under a in this longer edge. This is not true for b. Conversely, it is clear that if  z_4, z_5 are equal then swapping the lengths has no effect.
			
			This situation persists for the remaining gluing datums in this example.
			
			
			
  * Fourth case: if y_3 = y_1 + y_2 + y_4, then Figures <ref> and <ref> show realizations of . The edge t_4 was contracted. The drawing shows two different gluing datums (which reflects a bifurcation in the shrinking and regrowing sequence so far), but by swapping some branches one can see them to be isomorphic, so we label both by M_4.
			This observation applies as well to the remaining cases.
			
			 
				
  * Case 4.a
				
				
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						T_4 a			
					
					[t].32 
						M_4			
					
					[t].25 
						A_4 

					
					figure 
				
				

				
				
				
  * Case 4.b
				
				
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					[t].32
						T_4 b			
					
					[t].32 
						M_4			
					
					[t].25 
						A_4 

					
					figure 
				
				

			
			
			
			
  * Fifth case:
			
			 
				
  * Case 5.a: if y_1 + y_2 + y_4 < y_3 < y_1 + 2y_2 + y_4, then M_5 in Figure <ref> realizes .
				
				
				

				
				
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					[t].32
						T_5 a			
					
					[t].32 
						M_5			
					
					[t].25 
						A_5 = 16
					
					figure 
				
				

				
				
  * Case 5.b: if y_1 + y_2 + y_4 < y_3 < 2y_1 + y_2 + y_4, then M_5 in Figure <ref> realizes .
				
				

				
				
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					[t].32
						T_5 b			
					
					[t].32 
						M_5			
					
					[t].25 
						A_5 

					
					figure 
				
				

				
			
			
			
			
  * Sixth case:
			
			 
				
  * Case 6.a: if y_3 = y_1 + 2y_2 + y_4, then M_6 in Figure <ref> realizes . The edge t_2 was contracted.
				
				
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					[t].32
						T_6 a			
					
					[t].32 
						M_6			
					
					[t].25 
						A_6 
					
					figure 
				
				

				
				
  * Case 6.b: if y_3 = 2y_1 + y_2 + y_4, then M_6 in Figure <ref> realizes . The edge t_3 was contracted.
				
				
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					[t].32
						T_6 b			
					
					[t].32 
						M_6			
					
					[t].25 
						A_6
					
					figure 
				
				

				
			
			
			
			
  * Seventh case:
			
			 
				
  * Case 7.a: if y_1 + 2y_2 + y_4 < y_3 < y_1 + 2y_2 + y_4 + y_6, then M_7 in Figure <ref> realizes . To obtain this gluing datum from 6.a we have swapped branch copies 2 and 3 in the branch above t_6, and then regrown t_2.
				
				

				
				
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					[t].32
						T_7 a			
					
					[t].32 
						M_7			
					
					[t].25 
						A_7 = -16
					
					figure 
				
				

				
				
				
  * Case 7.b: if 2y_1 + y_2 + y_4 < y_3 < 2y_1 + y_2 + y_4 + y_8, then M_7 in Figure <ref> realizes . This was obtained from 6.b by swapping branch copies 2 and 3 in the branch above t_9, and then regrowing t_2. 
				
				
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					[t].32
						T_7 b			
					
					[t].32 
						M_7			
					
					[t].25 
						A_7 
					
					figure 
				
				

			
			
			
			
			
			
  * Eighth case:
			
			 
				
  * Case 8.a: if y_3 = y_1 + 2y_2 + y_4 + y_6, then M_8 in Figure <ref> realizes . The edge t_6 was contracted.
				
				
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					[t].32
						T_8 a			
					
					[t].32 
						M_8			
					
					[t].25 
						A_8
					
					figure 
				
				

				
				
  * Case 8.b: if y_3 = 2y_1 + y_2 + y_4 + y_8, then M_8 in Figure <ref> realizes . The edge t_8 was contracted.
				
				
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					[t].32
						T_8 b			
					
					[t].32 
						M_8			
					
					[t].25 
						A_8 
					
					figure 
				
				

			
			
			
			
			
  * Ninth case:
			
			 
				
  * Case 9.a: if y_1 + 2y_2 + y_4 + y_6 < y_3, then M_9 in Figure <ref> realizes .
				
				

				
				
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					[t].32
						T_9 a			
					
					[t].32 
						M_9			
					
					[t].25 
						A_9 = 16
					
					figure 
				
				

				
				
  * Case 9.b: if 2y_1 + y_2 + y_4 + y_8 < y_3, then M_9 in Figure <ref> realizes .
				
				

				
				
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					[t].32
						T_9 b			
					
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§ DEFORMATION
 
	We visualize Example <ref> as a movie featuring a continuous deformation of the length function ℓ_H (for an actual movie go here[https://mathsites.unibe.ch/jdraisma/MovieGenus4.mp4]). 
	The deformation path grows the length y_3 while leaving fixed the remaining lengths. The path moves in and through nine different
	cones of sources C_M. We use the full-rank property of A_M to calculate ℓ_T =  A_M ℓ_H, which defines the tropical morphism (M,ℓ_T). As we grow the length y_3, some lengths of ℓ_T shrink down to zero. We call the cases where a length of ℓ_T is zero a limit in the deformation. Since we do not allow zero lengths in an edge-length function, we contract an edge of T instead. 
	
	If we were to walk further past a limit while using the same gluing datum we would get negative entries in A_M ℓ_H, which is not allowed in a length function. In order to go beyond we pass to a different gluing datum. In Example <ref> the even cases are limit gluing datums, as an edge is contracted. Around these limits there is a change in the gluing relations. In this section we show that understanding this change of gluing relations sets up a deformation procedure and gives a proof of Theorem <ref>. 
	
	
	
	

 §.§ Limits
 
	We set up a framework to study limits. 
	
	Let M = (T, d, ∼) be a gluing datum, and t an edge of T.
	
	[Limit gluing datum]   The limit of M by contracting t, in short the limit at t, is the gluing datum M_0 = (T_0, d, ∼_0) given by the following data: 
		 
			
  * T_0 is obtained by contracting t in T. The ends u, v of t get identified with a vertex w_0 of T_0. Edges and vertices of T different from u, v, and t correspond in a natural manner with edges and vertices of T_0.  
			
  * ∼_0,w_0 equals the finest common coarsening of ∼_u and ∼_v; and ∼_0,x equals ∼_x for x  w_0.
		
		
	
	
	For the remainder of this section, let M_0 = (T_0, d, ∼_0) be the limit at t of M, giving rise to G_0, and u, v be the ends of t that contract to w_0 in V(T_0).
	Fix an edge t of T. Let _0 : G_0 → T_0 be the that arises from _M : G → T by contracting t in T to obtain T_0, and contracting all the edges in (t) to obtain G_0. It is straightforward to see that _0 is canonically isomorphic to _M_0, where M_0 is the limit of M at t. Namely, the classes representing vertices and edges of G contract to the classes representing vertices and edges of G_0.
	Note that a class x i of G contracts to x_0 j of G_0 if and only if x i ⊆x_0 j  and x contracts to x_0.
	
	To prove that M_0 is indeed a gluing datum, we observe that verticality, refinement and connectedness are inherited.  We write r_0 for r__0. The RH-condition amounts to proving that r_0(A_0) ≥ 0 for all A_0 in V(G_0). This is true because	 r_0 is additive under contraction: 
	
	
	 
		Let A_0 be in _0(w_0), and A_1, …, A_r be the vertices of G that contract to A_0. If g(H(M)) = g(H(M_0)), then: 
		
    r_0(A_0)    = r_φ(A_1) + … + r_φ(A_r), 
    _M_0 w_0    = _M u + _M v.

	
	 
		Let G_A be the subgraph defined by the vertices and edges of G that contract to A_0. Namely, V(G_A) = {A_1, …, A_r}. It is a forest; otherwise we would have g(H(M_0)) < g(H(M)); and also connected, hence a tree. Restrict  to G_A to get a map _A onto the graph on {u, v}, joined by the single edge t. This map is a of degree |A_0|. 
		By Lemma <ref> the total change of _A is ∑_B ∈ V(G_A) r__A (B) = 2|A_0| - 2. Using Lemma <ref> we calculate:
		
    ∑_i=1^r r_φ(A_i)    = ∑_i = 1^r (2|A_i| - 2)
    		
    		- ∑_i=1^r ∑_e ∈ E(A_i) 
    (e)  t (|e| - 1) 
    		- 2 ·∑_e ∈ E(G_A) (|e| - 1)
       = ∑_B ∈ V(G_A) r__A (B)
    		- ∑_i=1^r ∑_e ∈ E(A_i) 
    (e)  t (|e| - 1) 
       = 2|A_0| - 2 - ∑_e ∈ E(A_0) (|e| - 1) = r_0(A).

		Apply this formula over all vertices above w_0 to get the second result.
	
	
	

 §.§ Inherited properties
  
	
	Now we study which properties of Definition <ref> are inherited by limits. Let M_0, G_0, u, v, w_0, _0 and r_0 be as in Subsection <ref>.
	
	
	[dangling in the limit]  
		If g(H(M)) = g(H(M_0)), then a vertex A_0 in _0(w_0) is dangling if and only if all the A_1, …, A_r in V(G) that contract to A_0 are dangling. 	
	
	
	
	
		Since g(H(M)) = g(H(M_0)), no cycle of G is contracted in G_0. This implies that A_0 is not contained in a cycle of G_0 if and only if none of A_1, …, A_r are contained in a cycle of G.
	
	
	
	 
		If g(H(M))=g(H(M_0)) and M satisfies the dangling-no-glue condition, then M_0 satisfies the dangling-no-glue condition.
	
	
	
		We only need to check that a dangling vertex A_0 above w_0 satisfies |A_0| = 1. Since ∼_w_0 is the finest common coarsening of ∼_u and ∼_v, we have |A_0| > 1 only if for some A in V(G) contracting to A_0 we have |A|>1. By Lemma <ref> there is no such A.
	
	
	[non-dangling union]  Assume that g(H(M))=g(H(M_0)). Let M satisfy the dangling-no-glue condition, let A_0 in _0(w_0) be non-dangling, and let A_1, …, A_r in V(G) be the non-dangling vertices that contract to A. Then A_0 = A_1 ∪…∪ A_r, as subsets of [d].
	
	
	
		This is equivalent to proving that for every k in A_0 at least one of B_u =  u k, B_v =  v k in V(G) is non-dangling. Assume not, so by dangling-no-glue |B_u|=|B_v|=1, thus B_u, B_v are not incident to other vertices above u or v. Hence these two dangling classes are the only ones that contract to A_0. By Lemma <ref> A_0 is dangling, a contradiction.
	
	
	Note that in the previous lemmas we have used the fact that the gluing relations of M_0 agree with those of M outside of w_0. From this fact we also get that if M has an edge labelling λ = (λ_T, λ_H), with λ_T: E(T) → [|E(T)|] and λ_H: E(H(M)) → [|E(H(M))|] (recall that [m] means {1,…,m}), then they induce edge labellings λ_T,0 and λ_H,0 on M_0. It is clear how to do this for T_0. For calculating λ_H,0(h_0), with h_0 in E(H(M_0)), choose any e_0 ∈ h_0. As edges of G not above u, v, or t correspond to the edges of G_0, there is an edge e in G that corresponds to e_0. Recall that h(e) is the unique edge of H(M) containing e. Set λ_H,0(h_0) = λ_H(h(e)). It is straightforward to see that this is well defined.
	
	Going one step further, if two gluing datums share a limit, then loosely speaking  their combinatorial information agrees outside the contracted edge and its ends. We define a condition on the edge orders that allows us to relate the edge-length matrices. Recall from the proof of Lemma <ref> that an isomorphism (τ, π_x) from M to some gluing datum M' gives rise to a graph isomorphism γ: G → G'. This γ descends to a graph isomorphism : H(M) → H(M').
	
	
		Let M q = ( T q, d, ∼ q) be gluing datums with edge labellings λ_T q : E( T q) → [|E( T q)|] and λ_H q : E(H( M q)) → [|E(H( M q))|], for q=1,2. Let M_0 q be the limit at t_k = (λ_T q)(k) of M q, with λ_T,0 q and λ_H,0 q the induced labellings. If |E(H(M_0))| ≥ |E(H( M q))| - 1, then we say that the labellings of M 1 and M 2 are compatible at t_k if there is an isomorphism (τ, π_x) from M 1 _0 to M 2 _0 such that λ_T,0 2 = λ_T,0 1 ∘τ and λ_H,0 2 = λ_H,0 1 ∘.
	
	
	The condition |E(H(M_0))| ≥ |E(H( M q))| - 1 ensures that in M_0 at most one edge of H( M q) is contracted. This is needed to get unicity at the end of Remark <ref>.
	
	
	[limit matrix]  
		Let M 1, M 2 be gluing datums with labellings compatible at t_k, and edge-length matrices (a_ij), (b_ij) respectively. Then a_i j = b_ij for j  k. 
	
	
	
	
	
		Restrict the edge-length map of M 1 to the subspace {x_k = 0} to get the edge-length map of M_0 1. This amounts to deleting the k-th column of (a_ij). The matrix of the edge-length map of M_0 1 coincides with that of the edge length-map of M_0 2 because of the isomorphism and compatibility of labellings. Since the matrix of the edge length map of M_0 2 is obtained by deleting the k-th column of (b_ij), we get the result.   
	
	
	
	  If M has full-rank, then every limit of M has full-rank. 
	
	
	
	  If M satisfies pass-once, then every limit of M satisfies pass-once. 
	
	
	
		This is a consequence of Lemma <ref> and Proposition <ref>.
	
	
	 
		Note that if the limit of M at t, and the limit of M' at t' are isomorphic, then a labelling λ on M induces a unique labelling λ' on M' such that the two labellings are compatible at t_k with k = λ_T(t). Indeed, induce λ_0 on M_0 from λ; the isomorphism from M_0 to M'_0 gives a labelling λ'_0 from λ_0; and there is a unique λ' that induces λ'_0 on M'_0 such that λ'_T'(t') = λ_T(t); and if edges h', h of H(M'), H(M), respectively, were contracted, then λ'_H'(h') = λ_H(h)
	
	
	
	  If M satisfies no-return and pass-once, then a limit M_0 satisfies no-return if either w_0 is a leaf or w_0 ≤ 2. 
	
	
	
	
		Suppose that A_0 in V(G_0), with r= A ≥ 2, violates no-return. Since M satisfies no-return, then A_0 is above w_0. If w_0 = 1 there is nothing to check, so assume w_0 is not a leaf. If r_0(A_0) ≤ 1, we get a contradiction with Lemma <ref> (r1 implies no-return). Since w_0 ≤ 2, we have that r_0(A_0) = 2. As A_0 violates no-return, all the non-dangling edges e_1, …, e_r incident to A_0 are above a single edge of T, thus |e_1| + … + |e_r| ≤ |A|, so |A| ≥ 2. Hence, r_0(A) = 2 = r - 2 + 2|A| - (|e_1| + … + |e_r|) ≥ r- 2 + |A| gives |A|=2 and r = 2. So there is an edge of H_0 passing above w_0 by going through A_0, and twice above t. Since w_0 is not a leaf, M_0 does not satisfy pass-once. On the other hand, Lemma <ref> implies that M_0 satisfies pass-once, a contradiction.
	
	
	The condition of having w_0 ≤ 2 when w_0 is not a leaf is satisfied when M is change-minimal and has full-rank; in particular it is satisfied by full-dimensional M. Indeed, Proposition <ref> and the equality w_0 =  u +  v - 2 together imply that
	
    C w_0 +  w_0 - 3 = ( u +  u - 3) + ( v +  v - 3) + 1 = 1.

	This means that w_0 is 0,1,2 or 3. The last value implies that w_0 is a leaf. Thus:
	
	
	   Let M be change-minimal with full-rank, and M_0 be a limit of M with g(H(M)) = g(H(M_0)). Then M_0 has full-rank and satisfies dangling-no-glue, no-return, and pass-once. 
	
	
	
		This is a consequence of Lemmas <ref>, <ref>, <ref>, <ref> and Equation (<ref>).
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

 §.§ Deformation
 
	We introduce the deformation procedure. Let ω be a deformation path in the interior of some cone C_H. The deformation procedure proves that upon encountering a limit while deforming a tropical morphism  along ω, as in Example <ref>, it is possible to continue further by changing the combinatorial structure of the tropical morphism. This means passing to another gluing datum. The majority of the proof of this fact is deferred to Section <ref>. Here we use Lemma <ref> to reduce the proof to the problem of constructing full-dimensional gluing datums with a given limit. 
	
	
	Recall that 𝔽𝔻_g is the set of full-dimensional whose sources have genus g. We abuse notation and write M in 𝔽𝔻_g for _M in 𝔽𝔻_g.
	Let M_0 be the limit at t_1 of some M in 𝔽𝔻_g.  Denote by M_0 the set of isomorphism classes of gluing datums in 𝔽𝔻_g that contract to a limit in the isomorphism class of M_0. By Remark <ref> a choice of an edge labelling on M induces an edge labelling compatible at t_1 on each element of M_0. 
	
	Since ω is contained in a cone C_H, namely our walk does not change the combinatorial type of the source,
	assume that H(M_0) and H(M) are isomorphic. Thus, C_M_0 is a subset of C_H(M).
	Denote by |ℓ_1 - ℓ_2| the euclidean distance in ^E(H). We wish to prove:
	
	  
		Let M_0 be the limit at t_1 of some full-dimensional gluing datum M, with H(M_0) isomorphic to H = H(M). Let y_0 ∈ C_M_0⊂ C_H. 
		There exists a constant ε(M_0, y_0)>0 such that if  is an element of C_H ∖ C_M_0 with |y_0 - | < ε(M_0, y_0), then there is  in M_0 with H() isomorphic to H and ∈ C_.
	
	
	Proposition <ref> is called deformation I because in the part II of this series of articles we enhance it to show that the number of distinct ways one can walk into a limit coincides with the number of ways one can walk out of it, under a suitable notion of multiplicity. This notion of multiplicity is suggested by the balancing condition in Section <ref> of this article.
	
	We compute ε(M_0, y_0) to have the property that if y ∈^E(H) is in a ball B centered at y_0 with radius ε(M_0, y_0) and the labellings of the elements of M_0 are compatible at t_1, then given = (, d, ∼̂) in M_0 the only obstruction for ∈ B to also be in C_ is that (t_1) is positive.
	For  in M_0 write  for A_, and let _0 = (y_0) ∈^E(). Denote by min^+ y the smallest positive entry of y.  Every element of the open ball B(_0, min^+ _0) ⊂^E() has positive entries, except possibly the one corresponding to t_1. By continuity of  we can choose a number ε(, M_0, y_0) > 0 such that the image under   of the open ball with centre y_0 and radius ε(, M_0, y_0) > 0 lies in B(_0, min^+ _0). 
	We define
	
    ε(M_0, y_0) = min_∈M_0ε(, M_0, y_0).
 This is well defined and positive since 𝔽𝔻_g is finite. By construction:
	
	 
		Let  be in M_0 and  in C_H. Set =  (). If |y_0 - | < ε(M_0, ℓ_0), then ∈ C_ if and only if (t_1) > 0.
	
	
	Now we compute a formula for (t_1), showing that its sign depends only on  M_0, , and crucially only on Â. Let = (â_ij). Append a column of zeroes at the beginning of the edge-length matrix of M_0 to obtain A_0 = (a_ij).  From Lemma <ref> (limit matrix) it follows that â_ij = a_ij for all i and j ≥ 2. Hence, the first row of the adjugate matrix (Â) depends only on M_0. Let (c_1, c_2, …, c_3g-3) be that first row.  Recall that the adjugate matrix satisfies:
	
    (Â) ·Â = ( Â) ·  I = ĉ· I,

	where ĉ = (Â).  This formula holds in general for gluing datums with square edge-length matrices. This gives that:
	
    ∗ĉ   = ∑_i = 1^3g-3 c_i â_i1,
    
    	∗∗
    	0    = ∑_i = 1^3g-3 c_i â_ij = ∑_i = 1^3g-3 c_i a_ij,  for  j ≥ 2.

	
	Moreover, since  is full-dimensional we have 0, so = 1/(). Thus, 
	
    L1(t_1) 
    	   =  1 ĉ∑_i = 1^3g-3 
    	c_i  (h_i).

	Hence, the sum depends only on ĉ,  and on the c_i (which depend on M_0). Write c =  A_M as well. Applying Equation (<ref>) to z(t_1) = ( A_M )(t_1) we get that 
	
    z(t_1) 
    	   =  1 c∑_i = 1^3g-3 
    	c_i  (h_i).

	Using Lemma <ref> and comparing signs we obtain:
	
	
	 
		Assume that |y_0 - | < ε(M_0, ℓ_0) and ∉C_M_0. If =  c, then ∈ C_ if and only if ∈ C_M. If ≠ c, then ∈ C_ if and only if ∉C_M. 
	
	
	
	
	
	
	
	
	
	
	
	So we would like to construct full-dimensional gluing datums with prescribed limit at t_1 and a prescribed sign for A_M.
	
	 
		If M is in M_0, the edge labelling on M_0 is compatible at t_1, and H(M_0) is isomorphic to H(M), then there is  in M_0 with A_M ≠ A_.
		
		
	
	
	Our constructive proof of Lemma <ref> is the main achievement of this article. This is carried out in Section <ref>. Proposition <ref> (deformation I) follows easily from Lemma <ref>:
	
	 If ∈ C_M, then we are done, just take = M. Otherwise ∉C_M. By Lemma <ref> there is a gluing datum  in M_0 with ≠ c. By Lemma <ref> we have that ∈ C_. 
	
	
	
	
	

 §.§ Proof of main result
  
	To close this section, we put together the deformation procedure of Proposition <ref> and the initial families of Theorem <ref> to derive a proof of Theorem <ref>.
	
	
	
	
	
	
	
	
	
	
	
	
	Our deformation machinery is designed to work with full-dimensional gluing datums, which implies even genus. The odd genus case follows from the usual trick of attaching a lollipop. For executing the trick we need a converse to Lemma <ref>.
	
	
	(loop and bridge)  
		Let M be change-minimal and full-rank. Let A in V(H(M)) be trivalent, incident to a bridge h_b and a loop h_l. Then h_b and h_l are above a path of T of length 2 that leads to a leaf. Moreover, they are the only edges of H above this path. 
		
		
		
		
	
	
	
	
		Regard A as a vertex of G. Let v=(A). Suppose that r_(A) = 0. By Lemma <ref> we can apply Proposition <ref> (local properties). By the case (r0-nd3) the three non-dangling edges incident to A are above three distinct edges of T. Since two of the edges of G incident to A are in h_l, we have that h_l passes above at least two leaves of T, contradicting Remark <ref> (change-minimal leaves). The possibility r_(A) = 2 is ruled out since A = 3. Thus, r_(A) = 1. The case (r1-nd3) implies that v is divalent and there is an edge t in E(T) incident to v such that
		there are non-dangling edges e_1 and e_2 in A above t. Both e_1,e_2 are in h_l, for otherwise there is a contradiction again. Hence, h_l passes twice above t, so by pass-once t leads to a leaf. The path is made by t and the other edge incident to v, and now we can apply Lemma <ref> to conclude uniqueness.
		
	
	
	Let M = (T, d, ∼) be a genus-g gluing datum. We add some boundary points to C_M ⊂ C_H(M), in a process akin to the one done in Section <ref> that added some boundary points to C_H, giving rise to C_H. For a given ℓ_T in ^E(T) contract all the edges of T for which ℓ_T(e) = 0 to get M_0 (this is independent of contraction order); 
	the pair (M, ℓ_T) refers to the tropical morphism (M_0, ℓ|_T_0). Denote by C_M the image under A_M of the points ℓ_T in ^E(T) such that the source of (M, ℓ_T) has genus g; namely points where the genus does not drop. 
	
	
		The cone C_M ⊂C_H(M) is closed in the topology that ^E(H(M)) induces on C_H(M).
	
	
	Let H be a genus-g combinatorial type. We denote by H the subset of gluing datums M in g such that H(M) is isomorphic to H. To prove Theorem <ref> we show that it is possible to cover C_H with the cones C_M of the M in H, by identifying C_H with points in _≥ 0^3g-3. Recall that an edge labelling (λ_T, λ_H) on M identifies C_M with a cone in _≥0^3g-3. Moreover, recall from Subsection <ref> that if M_0 has trivalent H(M_0) and is the limit at t_1 of some M in H, then a labelling compatible at t_1 has the property of embedding in _≥0^3g-3 the cones of two M, M̂ in M_0 in such a way that the intersection is the embedding of the cone of M_0. Thus, embed H into ^3g-3 by choosing labellings for the elements of H such that for every M_0 that has trivalent H(M_0) and is a limit at t_1 of some M in H, we have labellings compatible at t_1 for M_0.
	
	We now present the proof of Theorem <ref>. To realize a point y_1 in C_H we use Theorem <ref> to choose an initial point y^* in C_H realizable by a tropical morphism, and draw a straight line in _≥ 0^3g-3 between y_1 and y^* to use as a deformation path. We then walk along this line, applying Proposition <ref> as needed.  To ensure that the hypotheses of Proposition <ref> are met, we choose y^* to be in general position using the following standard argument.
	
	 
		Let S = ⋃_L ∈ℱ L ⊂ℝ^n be a countable union of affine linear spaces L of codimension at least 2, and y, y_1 be in ℝ^n. For every ε > 0 there is a point y^* in the radius-ε ball centred at  y such that the half-open line segment [y^*, y_1) does not intersect S. 
	
	
	
		Let S̅ = ⋃_L ∈ℱaff-span (y_1,L) ⊂ℝ^n. The codimension of  aff-span (y_1,L) is at least 1; as ℱ is countable, this implies that the ball B(y, ϵ) is not contained in S̅. So choose y^* in B(y, ε) ∖S̅.
	
	
	 Assume that g = 2g'. To realize a genus-g metric graph , delete the dangling trees of  to obtain the metric graph , and let H be a trivalent combinatorial type such that for some y_1 ∈C_H we have that = (H, y_1). By Theorem <ref> we have that H is non-empty. Moreover it is a finite set. So embed H into C_H via labellings compatible at t_1.
		
		For a gluing datum M in H, denote by 𝕃_k M the set of affine spaces in ℝ^3g-3 associated to limits of M where at least k edges have been contracted. This means, images under A_M of sets in ℝ^ |E(T)| defined by fixing k coordinates to be zero; the members of 𝕃_k M have codimension at least k.
		Let 
    Z_k = ⋃_M ∈ H𝕃_k M.
 
		
		Since H is finite, the set Z_k is closed in the topology that ℝ^3g-3 induces on C_H. Choose M in H, and using Lemma <ref> choose y^* in C_M ⊂_≥ 0^3g-3 such that the half open line segment σ = [y^*, y_1) does not intersect Z_2. Note that y^* is not in Z_1 since C_M ∩ Z_1 = ∅ and y^* ∈ C_M. Thus, σ intersects with Z_1 finitely many times, and these intersections are not in Z_2. Now simply walk from y^* to y_1, always staying inside of the embedding of H into _≥ 0^3g-3 by applying Proposition <ref>, if necessary, when arriving at one of the finitely many points σ∩ Z_1.
		
		Now assume that g is odd. Choose a point x in  and attach to it a loop via a bridge. The lengths of the bridge and loop are not important, hence the construction depends on one parameter (the placement of x). The resulting graph _b = (G_b, ℓ_b) has even genus g+1. As g is odd, ⌈ g/2 ⌉ + 1 equals ⌈ (g+1)/2 ⌉ + 1. So there is a tropical morphism : _b→ (T, ℓ_T) of degree  ⌈ g/2 ⌉ + 1.  By Lemma <ref>, the bridge and loop added to  are above two edges of T, and no other point of _b is above them. Make the desired morphism by deleting these two edges and everything above. 
		
		
	
	
	

§ CONSTRUCTIONS
 
	We carry out the constructions necessary to prove deformation. 
	
	This section is motivated by the following paramount observation.
	The gluing datum M_2 of Example <ref> is the limit at t_1 of M_1, M a _3, and M b _3 with edge labellings λ_1, λ a _3 and λ b _3.  There is an isomorphism from M a _3 to M b _3, but λ a _3 does not induce λ b _3; moreover M a _3 and M b _3 produce in total two non-isomorphic tropical morphisms realizing the same source . 
	Let A_1, A a_2, A b_3 be the edge-length matrices of these gluing datums. They satisfy: 
    A_1 +  A a_3 +  A b _3 = 0.
 We view this equation as a balancing condition, something ubiquitous in tropical geometry.  
	We generalize this relation with Equation (<ref>) in Subsection <ref>. 
	
	We construct several candidates to be in M_0,
	for M_0 the limit at t_1 of some M in 𝔽𝔻_g. We use Equation (<ref>) to prove that at least in one of these candidates the determinant of the edge-length matrix has the sign opposite to that of A_M; as needed in Lemma <ref>.
	
	

 §.§ Candidates to being full-dimensional
 
	Let M be in 𝔽𝔻_g and M_0 be the limit at t_1. Recall that giving a construction for M_0 is hard because having a full-rank edge-length matrix is a global condition. Instead, we pass to a larger class of gluing datums that contains M_0. In return we get conditions that can be checked locally, namely they only depend on the gluing relations above t_1 and its ends u, v.
	
	
	
	  Let g = 2g' and M be a genus-g gluing datum. We say that M is possibly full-dimensional if M is change-minimal, satisfies dangling-no-glue, no-return, and A_M is a (3g-3)×(3g-3)-matrix. 
	
	
	We denote by M_0 the set of isomorphism classes of possibly full-dimensional gluing datums that contract to a limit isomorphic to M_0. By Lemma <ref> we have that M_0 contains M_0. We index the elements of M_0 by denoting them Mq, with q=1,2, and so on. We write T q for the base tree of M q, G q for the graph that arises from M q, q for _ M q, r q for r_ q, m q for m_ q, and A q = ( a q _ij) for A_ M q. We also write T_0 for the base tree of M_0, G_0 for the graph that arises from M_0, _0 for _M_0, r_0 for r__0, m_0 for m__0, and A_0 = (a_ij) for the matrix we get by inserting the zero column at the beginning of A_M_0.
	
	Recall that by Equation (<ref>) on Page eq-C we have that w_0 is not change-minimal; more precisely, w_0 +  w_0 = 4. Our strategy to construct the elements of M_0 is to regrow w_0 back to an edge in such a way that the resulting gluing datum is change-minimal and satisfies dangling-no-glue and no-return. This is possible because, as advertised, these conditions depend solely on the gluing relations above t_1 and its ends, u and v. 
	
	
	
	
	
	
	

 §.§ A balancing condition
  On the other direction, note that since A q is a (3g-3)×(3g-3)-matrix we have that M q in M_0 is full-dimensional if only if A q  0. So we need a relation for the determinants.  This is where the balancing condition suggested by Example <ref> comes in:
	
	[balancing condition] 
		
		
		If M is in M_0 and H(M_0) is isomorphic to H(M),
		then there exists a finite sequence of triples ( M q, λ q, K_q), indexed by q, where the M q are in M_0 with edge labellings λ q compatible at t_1 and the K_q are positive integers, such that every isomorphism class in M_0 has a representative in the M q, and the following equality holds:
		
    ⋆∑_q = 1^r K_q ( A q) = 0.

	
	
	It is straightforward to see that the balancing condition of Lemma <ref> implies Lemma <ref>.
	
	
		Since M is in M_0, we have that M is in M_0. Then by the balancing condition we have some  in M_0 with A_ A_M and A_ 0. The latter equation implies that  is in M_0, as desired.
	
	
	We prove the balancing condition and construct M_0, simultaneously, with a case-by-case analysis. 
	For verifying Equation (<ref>) we rely on formulas that follow from Equations (<ref>) and (<ref>) on Page eq-ast.
	To shorten, we write c q for Aq. 
	Consider the adjugate matrix Aq, with columns indexed by edges of H( Mq) and rows indexed by edges of Tq.
	Let c_1, …, c_3g-3 be the first row of Aq. Recall that the c_i depend only on M_0. For an edge h in H( Mq) or H(M_0) we write c_h for c_λ q_H (h) or c_λ_0,H(h), respectively.  Recall that given a non-dangling e in Gq or G_0 we denote by h(e) the unique edge of H( Mq) or H(M_0), respectively, that contains e. If e is dangling, we let c_h(e) be zero. Fix a j in [3g-3]. We define:
	
    σ_0(J,  j)    =  ∑_e ∈ Jc_h(e)|e|,  
    	  for   J ⊆_M_0(t_j),
    
    	σ q(J,  j)    =  ∑_e ∈ Jc_h(e)|e|, 
    	  for   J ⊆_ Mq(t_j).
	
	
	For convenience we write σ_0(j) for σ_0(_M_0(t_j),  j) and σ q(j) for σ q(_ M q(t_j),  j). 
	
	 
		The following equalities hold: 
		
    σ q(j)    =  c q  if  j = 1,  otherwise σ q(j) = 0, 
    σ_0(J,  j)    = σ q(J,  j)  for  j 1  and any  J.

	
	
		The first equality follows from Equations (<ref>) and (<ref>), and the fact that: 
		
    σ q(J,  j) =  ∑_e ∈_ Mq(t_j)  c_h(e)|e|
    		= ∑_i = 1^3g-3 c_i  aq_ij.

		Note that if j  1 the edges in _M_0(t_j) correspond to those in _ Mq(t_j), which together with the facts that the c_i depend only on M_0 and a_ij q = a_ij for j  1, gives the second equality.
	
	
	Additionally, note that if J_1, J_2 are disjoint, then σ_0(J_1,j) + σ_0(J_2,j) = σ_0(J_1 ∪ J_2,j).
	
	

 §.§ Cases for constructing PStar(M0)
 
	The plan of action to construct M_0 has three steps:
	
		
  * Finding the possible base trees that contract to T_0.
		
  * Restricting the possibilities for the gluing relations ∼_u q, ∼_v q, ∼_t_1 q.
		
  * Giving a construction of a gluing datum together with edge labellings compatible at t_1 for each of the possibilities found in Step (2). 
	
	
	We deal with Step (1) on Subsection <ref>.  Step (2) is the most delicate one, as it proves that indeed we have representatives for all the elements of M_0. We are able to say some useful generalities in Subsections <ref> and <ref>, but the definitive arguments come from assuming certain features of the gluing relations of M_0 above w_0 and edges in w_0. Hence, we split the proof in cases. For Step (3) we refer the reader to the illustrations that accompany each case.
	
	
	
	
	
	
	
	We have, by Equation (<ref>) on Page eq-C, that w_0 +  w_0 = 4. Thus, w_0 is 1, 2, 3 or 4. In our setting the first option is not possible, because if w_0 were 1, then we would have u =1 and v = 2 for all gluing datums in M_0, including the full-dimensional M. So Lemma <ref> applied to M would imply that there is a loop above t_1, so the genus of H(M_0) would drop upon contraction of t_1, a contradiction. 
	
	Let A_0 be a vertex above w_0. If H(M_0) is isomorphic to H(M), then H(M_0) is trivalent. Hence, A_0 is 0, 2, or 3. Also, since w_0 ≤ 2 we have that r_0(A_0) is 0, 1 or 2.
	
	Our philosophy of division of cases is similar to the one for proving the local properties:
	
	
	the different cases for calculating M_0 are mainly determined by the values of w_0, r_0(A_0), and A_0. There are other minor case specific factors that come into play. This gives a total of 17 cases. See Figure <ref> for a map that eases navigating through the 15 main cases. The diagram does not show 2 auxiliary cases contained in Case {aux-r0}, which we treat in Subsection <ref>.
	
	
	
	
	
	
	
	

 §.§ Trees contracting to T0
  We describe the possibilities for T q. Assume that the edges of T_0 incident to w_0 are labelled with 2, …,  w_0 + 1. To construct a tree that contracts to T_0 choose disjoint sets S, S' whose union is {2, …,  w_0 + 1}. At most one of S, S' may be empty. Replace w_0 with an edge t_1 with ends u, v; the edges indexed by S are incident to u, and those indexed by S' are incident to v. We denote the resulting tree by T_S. For ease of notation we omit braces; for example we write T_2,3 for T_{2,3}, and T_∅ for the tree corresponding to empty S. Without loss of generality we assume that |S| ≤ |S'|, and in case of equality that 2 ∈ S. This gives a one-to-one correspondence between unordered partitions of {2, …,  w_0 + 1} into two parts and trees contracting to T_0. Since M q is change-minimal, u ≤ 3 and v ≤ 3, that is |S|, |S'| ≤ 2. 
	
	

 §.§ The graph ndGqA0
  
	Let  be the subgraph of non-dangling elements of G q that contract to A_0. Since g(H(M)) = g(H(M_0)), the graph  is a forest; moreover, by Lemma <ref> it is a tree. By dangling-no-glue, determining the gluing relations ∼_u q, ∼_v q, ∼_t_1 q is equivalent to determining A for each A in V() and the index m q(x) of each element x in .
	
	
	
	 
		
		Let M be a gluing datum, M_0 the limit at t_1, and A_0 above w_0. If g(H(M)) = g(H(M_0)), then:
		
    A_0 = ∑_A ∈ V() (A - 2) + 2.

	
	
	 
		Since  is a tree, |E()| = |V()|-1. So we have that:
		
    A_0   =  ∑_A ∈ V()A  - 2|E()|
       = ∑_A ∈ V()A - 2(|V()|-1)
       = ∑_A ∈ V() (A - 2) + 2.

	
	
	Let A be in V(). By Lemma <ref> if A_0 = 2, then A = 2. If A_0 = 3, then either A = 2 or A = 3; moreover, the latter case occurs exactly once. So when A is 3 we denote by A q the unique vertex of G q with non-dangling valency 3 that contracts to A_0.
	We make two further observations, one on the edge set and one on the vertex set of :
	
	
	
	
	
	
	
	  
		Let r be the number of vertices of G q that contract to A_0 and belong either to Case (r1-nd3) or Case (r2-nd2) of Proposition <ref> (local properties). Then |E()| ≤ r+1 ≤  r_0(A_0) + 1.
	
	
	
		Let A be in V(). By the local properties and the fact that all the edges of  are above t_1, we have that _ A is either 1 or 2. Thus,  is in fact a path, so the inner vertices have valency 2. Note that valency 2 implies that A belongs either to Case (r1-nd3) or Case (r2-nd2), so the result follows.
	
	
	 
		If r_0(A_0) ≤ 1, then the vertices of  are the ends above u, v of the edges e q, where e q corresponds to e in A_0.
	
	
	
		Let A be in V(). Since r_0(A_0) ≤ 1, we have that r q(A) ≤ 1. Thus, by the local properties A is not above a leaf. So A satisfies the no-return condition, which means that not all of the edges in A are above t_1, so at least one of them corresponds to one of the edges in A_0.
	
	
	
	
	
	
	

 §.§ Combinatorial type
   
	We can obtain H(M_0) from H(M) by contracting a set of edges. If the number of edges contracted is zero, then H(M_0) and H(M) are isomorphic. This is equivalent to having a trivalent H(M_0).  This is the setting of Lemma <ref> (balancing condition). Thus, we assume for the rest of this section that H(M_0) and H(M) are isomorphic. Since the edge labelling on M_0 is compatible at t_1, the identification is canonical.
	
	
	The case where H(M_0) and H(M) are not isomorphic is of interest as well, and we deal with it in Part II of this series of articles. It is by studying the M_0 of M_0 with non-trivalent H(M_0) that we show how it is possibly to walk between cones C_H with different combinatorial types H. We use this to prove that the space d g we alluded to in the sneak peak part of Subsection <ref> is connected. 
	
	

 §.§ Case-work
  Let A_0 be above w_0. Let e in G q contract to A_0. Let A_u, A_v be the ends of e, above u, v respectively. If e is dangling, then it does not contribute to σ q (1). So, by Lemma <ref> (dangling in the limit) we can assume that A_0 is non-dangling. We denote by J_r the set of those e contracting to A_0 such that rq(A_u) +  rq(A_v) = r. By the discussion in Subsection <ref> we have that r_0(A_0) ≤ 2, hence:
	
    A q = σ q (1) = σ q (J_0, 1) + σ q (J_1, 1) + σ q (J_2, 1).

	In each of the following cases we calculate σ q (J_0, 1), σ q (J_1, 1), and σ q (J_2, 1) separately. To avoid having several levels of subindices, we write c(e) for c_h(e).
	
	
	
	
	
	
	
	
	

  §.§.§ Case {aux-r0}
 
	We begin by showing that T q = T_ S q determines  when r_0(A_0) is 0. This is an auxiliary case for calculating σ q (J_0, 1). Since w_0 ≥ 2, by Lemma <ref> (r1 implies no-return) we have that A_0 satisfies the no-return condition. Thus, if A_0 = 2, then _0 is injective on A_0. If A_0 = 3, suppose that _0 is not injective on A_0, so ∑_e ∈A_0 |e| ≤ 2|A_0|. Hence, r_0(A_0) =  A_0 - 2 + 2|A_0| - ∑_e ∈A_0 |e| ≥ 1, a contradiction. 
	
	
		
  * Case {aux-r0-nd2}:
		
		
		Assume that A_0 is 2. Let e_α, e_β be the edges in A_0, above t_α and t_β respectively. By Lemma <ref> the vertices of  are the ends of e_α q and e_β q above u or v. If {α, β}⊂ S q, the ends of e_α q, e_β q are above u; since  is connected, they equal one vertex A in G q, which equals A_0 as subsets of [d] by Lemma <ref>. Similarly if {α, β}⊂S' q. Otherwise, one end A_u is above u, the other end A_v above v, so they are distinct. By connectivity of  there is one edge e' joining A_u, A_v. By Lemma <ref> and since r_0(A_0) = 0, the vertices A_u and A_v belong to  Case (r0-nd2) of the local properties. So, as subsets of [d], the classes e_α q, A_u, e', A_v, e_β q are equal; and h(e_α q) = h(e') = h(e_β q).
		
		
		
  * Case {aux-r0-nd3}: Assume that A_0 is 3. Let e_α, e_β, e_γ be the edges in A_0, above t_α, t_β, t_γ, respectively. Since |_0(A_0)| = 3 and max(| S q|,|S' q|) ≤ 2, both intersections S q ∩_0(A_0) and S' q ∩_0(A_0) are non-empty. One of these intersections is a singleton. Assume without loss of generality that the singleton is {α}. By Lemma <ref> there is at most one edge in , therefore at most two vertices. By Lemma <ref> the vertices of  are the end A_2 of e_α q above u, and the vertex A_3 that is the end of both e_β q and e_γ q above v. So A_2 and A_3 are distinct, joined by a non-dangling edge e', and A_2 = 2, A_3 = 3. By the Case (r0-nd2) of the local properties, as subsets of [d], we get that e_α q, A_2, and e' are equal, and h(e_α q) = h(e'). Note that they are also a subset of A_3. Finally Lemma <ref> implies that A_0 = A_3 ∪ A_2 = A_3.
	
	
	
	
	
	

  §.§.§ Case {w4}
  
	
	Assume that w_0 is 4. Then w_0 = 0 by Equation (<ref>) on Page eq-C, so r_0(A_0) = 0. 
	Thus, the two Cases {aux-r0-nd2} and {aux-r0-nd3} settle entirely this case. 
	
	Namely, the elements of M_0 are M 1, M 2 and M 3, determined by T_2,3, T_2,4, and T_2,5, respectively. 
	
	
	
	
	
	
	
	
	
		
  * Case {w4-r0-nd2}: Let A_0 = {e_α, e_β} with _0(e_α) = t_α, _0(e_β) = t_β. Applying Case {aux-r0-nd2} we get the local parts shown in Figure <ref>, and a formula.
		
		
		
			[t].2
				
				
				[scale=0.9]/250.pdf 
					(45,26)0.8w_0 
					(16,0) 0.8t_α
					(3,72) 0.8t_β
					(90,72)0.8t_γ 
					(75,0) 0.8t_δ		 
				
				
				

				
				M_0
			
			
			[t].22 
				/252.pdf 
					(47,34) 0.8t_1
					(31,34) 0.8u
					(64,34) 0.8v
					(9,9) 0.8t_α
					(0,62) 0.8t_β
					(96,62) 0.8t_γ
					(83,9) 0.8t_δ		
				 
				
				

				
				T_α,β
			
			
			[t].22
				/251.pdf 
					(47,34) 0.8t_1
					(31,34) 0.8u
					(64,34) 0.8v
					(9,9) 0.8t_α
					(0,62) 0.8t_γ
					(96,62) 0.8t_β
					(83,9) 0.8t_δ
				 
				
				

				
				T_α,γ
			
			
			[t].22 
				/251.pdf 
					(47,34) 0.8t_1
					(31,34) 0.8u
					(64,34) 0.8v
					(9,9) 0.8t_α
					(0,62) 0.8t_δ
					(96,62) 0.8t_β
					(83,9) 0.8t_γ
				
				
				
  
				
				T_α,δ
			
			figure 
			
		
		
		
    w4-nd2∑_q=1^3 ∑_e ∈ E()c(e)|e| = c(e_α) |e_α| + c(e_β)|e_β|.

		
		
  * Case {w4-r0-nd3}: Let A_0 = {e_α, e_β, e_γ} with _0(e_α) = t_α, _0(e_β) = t_β, _0(e_γ) = t_γ. Applying Case {aux-r0-nd3} we get the local parts shown in Figure <ref>, and a formula.
		
		

		
		
		
			[t].2
				
				
				[scale=0.9]/253.pdf 
					(45,26)0.8w_0 
					(16,0) 0.8t_α
					(3,72) 0.8t_β
					(90,72)0.8t_γ 
					(75,0) 0.8t_δ		 
				
				
				

				
				M_0
			
			
			[t].22 
				/254.pdf 
					(47,34) 0.8t_1
					(31,34) 0.8u
					(64,34) 0.8v
					(9,9) 0.8t_α
					(0,62) 0.8t_β
					(96,62) 0.8t_γ
					(83,9) 0.8t_δ		
				 
				
				

				
				T_α,β
			
			
			[t].22
				/254.pdf 
					(47,34) 0.8t_1
					(31,34) 0.8u
					(64,34) 0.8v
					(9,9) 0.8t_α
					(0,62) 0.8t_γ
					(96,62) 0.8t_β
					(83,9) 0.8t_δ
				 
				
				

				
				T_α,γ
			
			
			[t].22 
				/255.pdf 
					(47,34) 0.8t_1
					(31,34) 0.8u
					(64,34) 0.8v
					(9,9) 0.8t_α
					(0,62) 0.8t_δ
					(96,62) 0.8t_β
					(83,9) 0.8t_δ
				
				
				
  
				
				T_α,δ
			
			figure 
		
		
		
		
    w4-nd3∑_q=1^3 ∑_e ∈ E()c(e)|e| = c(e_α) |e_α| + c(e_β)|e_β| + c(e_γ)|e_γ|.

		
		
  * We verify Equation (<ref>) for Case {w4} by putting together Lemma <ref> and Equations (<ref>), (<ref>): 
		
    c 1 +  c 2 +  c 3 = ∑_q=1^3 σ q (1)    =  ∑_q=1^3  ∑_A_0 ∈_0(A) ∑_e ∈ E()c(e)|e|
       = ∑_j=2^5 σ q (j) = 0.

	
	
	
	
	
	
	

  §.§.§ Case {w3}
  
	
	
	Assume that w_0 is 3. Then w_0 = 1 by Equation (<ref>). There is exactly one vertex of G_0 with r_0-value 1 above w_0, the others have r_0-value 0. By Subsection <ref>, the possible base trees contracting to T_0 are T_2, T_3 and T_4; all with u = 2, v = 3. 
	
	Let {α} =  S q. We analyse the cases where r_0(A_0) is either 0 or 1.
	
	
		
		
  * Case {w3-r0}: Assume that r_0(A_0) = 0, and let A_u in G_A_0 q be non-dangling above u. Then r q(A_u) = 0. Since u = 2, only Case (r0-nd2) of the local properties is possible, so A = 2 and as a subset of [d] the class A_u is equal to both of the elements in A_u. Since any non-dangling edge in ( q)(t_α) has a non-dangling end above u, we conclude that:
		
    σ q(J_0,1) = σ q (J_0,α) = σ_0(J_0,α).
 
		
		
  * Case {w3-r1}: Since H_0 is trivalent either A_0 is 3 or 2.
		
		
  * Case {w3-r1-nd3}: Assume that  A_0 is 3. By the no-return condition _0(A_0) has at least two elements. So let e_2, e_3, e_4 be the edges in A_0, labelled such that |e_4| = max(|e_2|, |e_3|, |e_4|), _0(e_3) = t_3 and _0(e_4) = t_4. Let k_i = |e_i|. 
		Lemma <ref> (nd. r_ formula) gives r_0(A_0) = 1 = 3 + 2|A_0| - 2 - (k_2 + k_3 + k_4). That is, 2|A_0| = k_2 + k_3 + k_4. If e_2 were above t_4, then k_2 + k_4 ≤ |A_0|. Substituting in k_3 = 2|A_0| - (k_2 + k_4) gives k_3 ≥ |A_0| ≥ k_2 + k_4 > k_4, a contradiction. So e_2 is either above t_2 or above t_3.
		
		
		
  * Case {w3-r1-nd3-t2}: Assume that e_2 is above t_2.  
		Let {α, β, γ}={2,3,4}.
		Recall from Subsection <ref> that A q is the unique vertex of G q with A q = 3 that contracts to A_0.
		We say that M q is in Position I or II if A q is above u or v, respectively. We prove that T q and the position of A q determine M q.
		
		Position I: Assume that A q is above u. 
		
		Since u = 2 and A q = 3 we have that A q belongs to Case (r1-nd3) of the local properties (assuming Case (r0-nd3) contradicts that u = 2).
		So there are exactly two edges e', e” of A q above t_1; moreover |e_α| = | A q| = |e'| + |e”|. 
		By Lemma <ref> these are the two edges of . Hence,  has three vertices. By Lemma <ref> these are: A q incident to e_α q; and the two ends of e_β q, e_γ q above v, with non-dangling valency 2 and r q-value equal to 0. By the Case (r0-nd2) of the local properties we conclude that as subsets of [d], the edges e', e_β q and their common end are equal, and h(e') = h(e_β q); similarly with e” and e_γ q. Thus, |e_α| = |e_β| + |e_γ| and | A q| = |A_0|, so this construction is possible only when α = 4 and k_4 = |A_0|.
		
		Position II: Assume that A q is above v. 
		Since v = 3 and A q = 3 we have that A q belongs to Case (r0-nd3) of the local properties (assuming Case (r1-nd3) contradicts that v = 3). So there is exactly one edge e' of A q above t_1. 
		Let A' be the other end of e', which by Lemma <ref> has non-dangling valency 2. Lemma <ref> implies that the two vertices of  are A q and A'. Hence, by Lemma <ref> A' is incident to e_α q; and h(e_α q) = h(e'). Since r_0(A_0) = 1, we have that A' belongs to Case (r1-nd2) of the local properties and this gives two possible cases for the values of |A'|, |e'| and | A q|:
		
		Position II.a: |A'| = |e'| and |A'| = |e_α q| + 1. In this case, as subsets of [d], we get that e_α q ⊂ A', that A' and e' are equal, that e' ⊂ A q, and therefore, by Lemma <ref>, that A_0 =  A q ∪ A' =  A q. We also get that |A_0| ≥ |A'| > k_α.
		
		Position II.b: |A'| = |e'| + 1 and |A'| = |e_α q|. As subsets of [d] we have that e' ⊂ A' ∩ A q, so A' ∖ A q is either empty or a singleton. The former would imply that for i ∈ A' ∖ e', the classes e' and t_1 i _ M q are distinct edges of G q above t_1 with equal ends, which gives a cycle over t_1, a contradiction. So |A_0| = | A q| + |A'| - | A q ∩ A'| = | A q| + 1. The ends of e_β q and e_γ q are above v, so | A q| ≥ k_β, k_γ. Hence, |A_0| > max(k_β, k_γ), so k_α≥ 2.
		
		
		
		Since Position I implies |A_0| = k_4, we treat two cases: |A_0| = k_4 and |A_0| > k_4.
		
		
		
  * Case {w3-r1-nd3-t2-(a=k4)}: Assume that |A_0| = k_4. Then k_2 + k_3 = |A_0|, so k_4 > k_2, k_3. Let α = 4, so |A_0| = k_4, which precludes Position II.a. Position I and Position II.b give M 1 and M 2, respectively. Let α = 2, which precludes Position I. Then it is not true that |A_0| > max(k_β, k_γ) = k_4, which precludes Position II.b as well. Position II.a gives M 3. An analogous argument with α = 3 gives M 4. 
		See Figure <ref>.
		
		

		
		
		.3
			[scale=1]/145.pdf  
				(37,74) 1A_0
				(60,19) 1t_4
				(26,84) 1t_2
				(26,8) 1t_3
				(35,19) 1w_0
				
				(-2,93) 1e_2
				(-2,3) 1e_3 
				(83,48) 1e_4  
			
			
			M_0
		 
		
		.25
			 
    c(e_4)k_2 + k_3 + σ_0(J_0, 4)    = 0
    c(e_2)k_2 + σ_0(J_0, 2)    = 0
    c(e_3)k_3 + σ_0(J_0, 3)    = 0

			
		
		
		.33
			[scale=1]/148.pdf  
				(43,4) 1v
				(70,4) 1u
				(56,4) 1t_1
				(70,67) 1A^(1)  
			
			
			M^(1)
			
    |e'| = k_2      |e”| = k_3
    				
       σ q(J_0, 1) = σ_0(J_0, 4)
        c1 = c(e_2)k_2 + c(e_3)k_3 + σ q(J_0, 1)

			
		
		
		
		
			
			[t].3
				[scale=1]/149.pdf 
					(44,12) 1v
					(70,12) 1u
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					(43,75) 1A^(2)
					
					(14,93) 1e_2
					(-2,6) 1e_3 
					(101,44) 1e_4   
				
				
				M^(2) 
 

				|e'| = k_4 - 1 = k_2 + k_3 - 1
   σ 2(J_0, 1) = σ_0(J_0, 4)
   c^(2) = c(e_4)k_2 + k_3 - 1 + σ 2(J_0, 1)
				
			 
			
			[t].3
				[scale=1]/146.pdf
					(30,89) 1u
					(46,73) 1v
					(38,81) 1t_1
					(41,10) 1A^(3) 
				
				
				M^(3) 
 

				|e'| = k_2 + 1
					
   σ 3(J_0, 1) = σ_0(J_0, 2)
   c^(3) = c(e_2)k_2 + 1 + σ 3(J_0, 1)
				
			
			
			[t].3
				[scale=1]/147.pdf 
					(31,0) 1u
					(45,12) 1v
					(39,6) 1t_1
					(45,74) 1A^(4) 
				
				
				M^(4) 
 

				|e'| = k_3 + 1
   σ 4(J_0, 1) = σ_0(J_0, 3)
   c^(4) = c(e_3)k_3 + 1 + σ 4(J_0, 1)
				
			 
			figure 
		
		
		
		To verify Equation (<ref>) we compute: 
		
    c^(1) + (k_2 + k_3 - 1)c^(2) +(k_2+1) c^(3) + (k_3+1) c^(4)
       = ( (k_2 + 1)k_2c(e_2) + (k_2 + 1)σ_0(J_0, 2) ) +
    		((k_3 + 1)k_3c(e_3) + (k_3 + 1)σ_0(J_0, 3) )
       + (c(e_4) + (k_2 + k_3)σ_0(J_0, 4)  )
    		= 0 + 0 + 0 = 0.

		
		

		
		
		
  * Case {w3-r1-nd3-t2-(a>k4)}: Assume that |A_0| > k_4, which precludes Position I. If min(k_2, k_3, k_4) were 1, then k_2 + k_3 + k_4 ≤ 2|A_0| - 1, contradicting 0 = 2|A| - k_2 - k_3 - k_4. Hence, min(k_2, k_3, k_4) ≥ 2, so both cases of Position II are possible. It turns out that they balance each other for each possible α. The case α = 4 is shown in Figure <ref>. 
		
 
		
		
		
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					(60,18) 1t_4
					(26,82) 1t_2
					(26,8) 1t_3
					(34,18) 1w_0
					(36,70) 1A_0
					
					(-2,79) 1e_2
					(-2,10) 1e_3 
					(92,51) 1e_4  
				
				
				M_0
			
			
			.3
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					(47,18) 1t_1
					(60,18) 1u
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					(36,70) 1A^(2)  
				
				
				M^(2)
			

			
			.3
				c(e_4)k_4 + σ_0(J_0, 4) = 0
				
			
			
			.3
				|e'| = k_4 - 1
   σ 1(J_0, 1) = σ_0(J_0, 4)
   c^(1) = c(e_4)k_4-1 + σ 1(J_0, 1)
				
			
			
			.3
				|e'| = k_4 + 1
   σ 2(J_0, 1) = σ_0(J_0, 4)
   c^(2) = c(e_4)k_4+1 + σ 2(J_0, 1)
				
			
			figure 
		
		
		
		We conclude that:
		
    (k_4-1)c^(1) + (k_4+1)c^(2) 
    		= 2c(e_4) + 2k_4σ_0(J_0, 4) = 0.

		
		
		Analogous constructions exist for α = 2, which gives M 3, M 4; and α = 3, which gives M 5, M 6. 
		We verify Equation (<ref>):
		
    ( (k_4-1)c^(1) + (k_4+1)c^(2)) + 
    		( (k_2-1)c^(3) + (k_2+1)c^(4)) + 
       ( (k_3-1)c^(5) + (k_3+1)c^(6)) = 0 + 0 + 0 = 0.

		
		

		
		
		
  * Case {w3-r1-nd3-t3}: Assume that e_2 is above t_3. Then k_2 + k_3 ≤ |A_0|. Recall that 2|A_0| = k_2 + k_3 + k_4 and |A_0| ≥ k_i, so k_4 = |A_0|, k_2 + k_3 = |A_0| and |A_0| > k_2, k_3. We argue that T q determines . If α were 2, then by Lemma <ref> all the vertices of  would be above v, so  would have a single vertex with r q-value 1 above trivalent v, contradicting that M q is change-minimal. For α = 3,4, we get that  has vertices above u and v. The vertices above v belong to the Case (r0-nd2) of the local properties, since Case (r0-nd3) would produce a non-dangling element above t_2. So if α = 3, we get M 1 where e_2 q, e_3 q have ends above v, and by Case (r0-nd2) these determine the two edges e', e” of ; similarly, if α = 4 we get M 2 where e_4 q has an end above v and this determines the edge e' of . So the vertex of  above u is A q, and it belongs to Case (r1-nd3). So A q equals e_4, equals A_0 as subsets of [d]. 
		
		
		
		
		
		
		
		
		
		
		This gives M 1 and M 2 for α = 3,4, respectively. See Figure <ref>.
		

		
		
		
			[t].3
				[scale=1]/284.pdf  
					(25,9) 1t_3
					(62,9) 1t_4
					(46,9) 1w_0
					(-5,44) 1e_2	
					(-5,21) 1e_3	
					(98,34) 1e_4	 
					(46,55) 1A_0
				
				
				
				M_0 
 

				c(e_2)k_2  + c(e_3)k_3 + σ_0(J_0, 3) = 0
   c(e_4)k_2 + k_3  + σ_0(J_0, 4) = 0  
			
			
			[t].3 
				[scale=1]/285.pdf   
					(42,10) 1t_1
					(33,10) 1u
					(57,10) 1v
					(34,55) 1A^(1)
				  
				
				M 1 
 
 |e'| = k_4 = k_2 + k_3
   σ 1(J_0, 1) = σ_0(J_0, 3)
    c 1 = c(e_4)k_2 + k_3 + σ 1(J_0, 1) 
			
			
			[t].3 
				[scale=1]/286.pdf  
					(45,10) 1t_1
					(35,10) 1v
					(57,10) 1u
					(56,55) 1A^(2)
				
				
				M 2 
 
 |e'| = k_2      |e”| = k_3
   σ 2(J_0, 1) = σ_0(J_0, 4)
    c2 = c(e_2)k_2  + c(e_3)k_3
       +  σ 2(J_0, 1)  
			
			figure 
		
		
		
		
		We verify Equation (<ref>):
		
    c^(1) + c^(2) = 
       (  1 k_2 + k_3  c(e_4) + σ_0(J_0, 4) ) +
    		(  1 k_2 c(e_2) +  1 k_3 c(e_3) + σ_0(J_0, 3) )  = 0 + 0 = 0.

		
		

		
		
		
  * Case {w3-r1-nd2}: Assume that A_0 is 2. Let e, e' be in A_0. They are in the same edge h of H_0. Assume that |e| ≤ |e'|. By no-return we may assume that e and e' are above t_3 and t_4 respectively. Lemma <ref> (nd. r_ formula) gives r_φ(A_0) = 1 = 2|A_0| - |e'| - |e|. Thus, if k=|e|, then |A_0| = |e'| = k + 1. From here, follow a  reasoning analogous to case {w3-nd3-t2} to show that T q determines M q, and that the vertices of  above v belong to Case (r0-nd2) of the local properties, and the ones above u to Case (r1-nd2). See Figure <ref>.
		

		
		
		
			[t].3 
				[scale=1]/287.pdf 
					(25,9) 1t_3
					(62,9) 1t_4
					(46,9) 1w_0
					(-2,20) 1e
					(98, 27) 1e'	
					(46,43) 1A_0
				
				
				M_0
				
    c_hk  + σ_0(J_0, 3) = 0
       c_hk+1 + σ_0(J_0, 4) = 0

			
			
			[t].3
				[scale=1]/288.pdf  
					(45,10) 1t_1
					(35,10) 1u
					(57,10) 1v
					
				
				
				M^(1)
				
    σ 1(J_0, 1) = σ_0(J_0, 3)
        c1 = c_hk+1 + σ 1(J_0, 1)

			
			
			[t].30 
				[scale=1]/289.pdf 
					(45,10) 1t_1
					(35,10) 1v
					(57,10) 1u
					
				
				
				M^(2)
				
    σ 2(J_0, 1) = σ_0(J_0, 4)
        c 2 = c_hk +  σ 2(J_0, 1)
 
			
			figure 
		
		
		
		We verify Equation (<ref>):
		
    c^(1) + c^(2) = 
    		 (  1k c_h + σ_0(J_0, 3)) +
    		 (1k+1 c_h + σ_0(J_0, 4))  = 0 + 0 = 0.

		
		
	
	
	
	
	
	

  §.§.§ Case {w2}
  
	
	Assume that w_0 is 2. Then r_0(A_0) =  A_0 - 2. So if r_0(A_0) = 0, then A_0 = 2. Hence, σ_0 (J_0, 2) = σ_0 (J_0, 3), and we denote this quantity by s. Also,  w_0 = 2 by Equation (<ref>). 
	The trees contracting to T_0 are T_∅ with u = 1, v = 3; and T_2 with u =  v = 2. If T q = T_∅, then Remark <ref> implies that σ q(J_0, 1) = σ q (J_1, 1) = 0, so A q = σ q (J_2, 1). If T q = T_2, then a reasoning analogous to Case {w3-r0} gives
	
    σ q(J_0, 1)    = σ q (J_0, 2) = s, 
    σ q(J_0, 1)    = σ q (J_0, 3) = s.
 
	So we are left with the cases r_0(A_0) = 2 and r_0(A_0) = 1.
	
	 
		
		
  * Case {w2-r2}: Assume that r_0(A_0) = 2. Then A_0 = 4. Since A_0 satisfies no-return, let e_2, e_3 be non-dangling edges in A_0, above t_2, t_3, respectively. Let e_1, e_4 be the remaining two edges of A_0. We may assume without loss of generality that e_1 is above t_2. By the refinement property:
		
    ∑_e ∈A_0
    _0(e) = t_2 |e| = ∑_e ∈A_0
    _0(e) = t_3 |e| = |A_0|.

		So in particular ∑_i=1^4 k_i = 2|A_0|. We explore how T q affects :
		
		Base I: assume that T q is T_∅. Then, by Remark <ref> (change-minimal leaves) we have that  has a vertex A_u above u with |A_u| = 2, r q(A_u) = 2, and two incident edges e', e” with |e'| = |e”| = 1.  These are all the edges of  by Lemma <ref> and the fact that r_0(A_0) = 2. Thus, A_u and the ends A', A” of e', e”, respectively, are the vertices of .  We may assume that e_2 q and A' are incident. There are two possibilities for A' and A”. Either e_3 q is incident to A' or A”.  We call the first case Base I.a. Since r q (A') =  r q(A”) = 0, we have that k_2 = |A'| = k_3, k_1 = |A”| = k_4. We call the second case Base I.b; it implies that k_2 equals the cardinality of the edge above t_3 incident to A'. So either k_2 = k_4 and k_3 = k_1; or k_2 = k_1 and k_3 = k_4. In particular, here the two edges of A_0 are above t_2, and the other two above t_3. See Case {w2-r2-nd3-M-11} for figures of both possibilities.
		
		Base II: assume that T q is T_2. Then the vertices of  belong to Cases (r0-nd2), (r1-nd2), or (r1-nd3) of the local properties. By Lemma <ref> at most one vertex of  belongs to Case (r1-nd3). So by Lemma <ref> we get that  has either one edge e'; or two edges e', e”. We call them Base II.1, Base II.2, respectively. 
		
		
		
		
		
  * Case {w2-r2-nd3}: Assume that A_0 is 3. We may assume without loss of generality that e_4 is dangling. So k_4 = 1 and 2|A_0| - 1= ∑_i=1^3 k_i. If e_4 is above t_3, then k_3 + 1 = |A_0| = k_1 + k_2. If e_4 is above t_2, then k_3 = |A_0| = k_1 + k_2 + 1. 
		
		
		
		We call these two possibilities Cardinality M and Cardinality P.
		
		For Base II we have that exactly one vertex of  belongs to Case (r1-nd3), so the other vertex of  with r q-value 1 belongs to Case (r1-nd2). 
		
		In Base II.1, Lemma <ref> implies that the ends of e_1 q, e_2 q above u are the same vertex of ; by no-return this vertex has non-dangling valency 3, so it is A q. By Case (r1-nd3) we have that e_1 ∪ e_2, A q, and e' are equal as subsets of [d]; therefore, |e'|=| A q| = k_1 + k_2. Hence, the other end A' of e' (which is above v, incident to e_3 q) belongs to Case (r1-nd2). Either |e'| = |A'| = k_3 + 1, or |e'| = |A'|-1 = k_3 - 1. The first possibility implies k_1 + k_2 = k_3 + 1, namely Cardinality M; the second possibility implies k_1 + k_2 = k_3 - 1, namely Cardinality P. We call these possibilities Base II.1.M and Base II.1.P, respectively.
		
		In Base II.2,  has three vertices. By Lemma <ref> these are the ends A', A” of e_1 q, e_2 q, respectively, above u,  and the end B of e_3 q above v. By no-return we may assume that e', e” are incident to A', A”, respectively. Since these are all the edges of , A' =  A” = 2, and B = 3. So B is A q and h(e_1 q) = h(e'), h(e_2 q) = h(e”). 
		So Case (r1-nd3) gives that k_3 = | A q| = |e'| + |e”|. One of A', A” belongs to Case (r1-nd2) and the other to Case (r0-nd2). If A' belongs to Case (r1-nd2) then either |e'| = |A'|-1 = k_1 - 1, so k_1 > 1 and (k_1-1) + k_2 = k_3, that is Cardinality M; or |e'| = |A'|+1 = k_1 + 1, which gives (k_1+1) + k_2 = k_3, that is Cardinality P. We call these possibilities Base II.2.1.M and Base II.2.1.P, respectively. Analogously, if A” belongs to Case (r1-nd2) we get Base II.2.2.M and Base II.2.2.P, where the former gives k_2 > 1.
		
		
		
		
  * Case {w2-r2-nd3-M}: Assume that e_4 is above t_3, namely Cardinality M. The possibilities are Base I.a, Base I.b, Base II.1.M, Base II.2.1.M, and Base II.2.2.M.
		
		
		
  * Case {w2-r2-nd3-M-11}:  Assume that k_1 = 1 and k_2 = 1. Base II.2.1.M, Base II.2.2.M are precluded since k_1 ≯1, k_2≯1, respectively. Base I.a, Base I.b, Base II.1.M determine M 1, M 2 and M 3, respectively. See Figure <ref>.
		
		
		.45
			[scale=1.3]/290.pdf 
					(51,29) 1A_0
					(35,4) 1t_2
					(67,4) 1t_3
					(51,4) 1w_0
					(-3,25) 1e_1	
					
					(-3,10) 1e_2	
					(100,10) 1e_3	 
			 
			
			M_0
		
		0.45
			
    c(e_1) + c(e_2) + s= 0
       c(e_1) + s = 0

		
		
		
 
		
		
			[t].30
				[scale=1]/291.pdf  
					(47,16) 1t_1
					(52,42) 1A^(1) 
					(39,4) 1u 
					(53,31) 1v 
					(-5,53) 1e_1	
					(-5,38) 1e_2	
					(98,38) 1e_3   
				
				
				M^(1) 
 

				σ 1(J_0, 1) = σ 1(J_1, 1) = 0
    c1 = 2c(e_1)
				
			
			
			[t].30 
				[scale=1]/292.pdf     
					(47,16) 1t_1
					(48,58) 1A^(2)
					(39,4) 1u 
					(53,31) 1v 
				 
				
				M^(2) 
 

				
					σ 2(J_0, 1) = σ 2(J_1, 1) = 0
    c2 = 2c(e_2)
				
			
			
			[t].30 
				[scale=1]/293.pdf       
					(48,31) 1t_1
					(37,31) 1u 
					(59,31) 1v 
					(36,58) 1A^(3)
				
				
				M^(3) 
 

				|e| = 2
   σ 3(J_0, 1) = s
   c^(3) = c(e_3) 2 +  s 
				
			
			figure 
		
		
		
		We verify Equation (<ref>):
		
    c^(1) + c^(2) + 4 c^(3) = 2(c(e_1) + c(e_2) + s) + 2(c(e_3) + s) = 2 · 0 + 2·0 = 0.

		
		

		
		
		
  * Case {w2-r2-nd3-M-1k}: Assume that k_1 = 1 and k_2 ≥ 2. Let k = k_2, so |A_0| = k + 1 and |e_3|=k. Base I.b and Base II.2.1.M are precluded since |e_2||e_4| and k_1≯1, respectively. Base I.a, Base II.2.2.M, Base II.1.M determine M 1, M 2 and M 3, respectively. See Figure <ref>.
		
		

		
		
		.5
			[scale=1.3]/294.pdf   
				(33,-3) 1t_2
				(67,-3) 1t_3
				(48,-3) 1w_0
				(-3,37) 1e_1	
				(-3,13) 1e_2	
				(98,13) 1e_3	 
				(50,42) 1A_0
			 
			

			
			M_0
		
		0.45
			
    c(e_1) + c(e_2) k + s = 0
       c(e_3) k + s = 0

		  
		
		
 
		
		
		
			[t].30
				[scale=1]/295.pdf 
					(47,15) 1t_1
					(52,61) 1A^(1) 
					(39,2) 1u 
					(53,29) 1v 
					(-5,69) 1e_1	
					(-5,45) 1e_2	
					(98,45) 1e_3  
				
				
				M^(1) 
 

				σ 1(J_0, 1) = σ 1(J_1, 1) = 0
    c 1 = 2c(e_1)  
			 
			
			[t].3 
				[scale=1]/296.pdf    
					(51,28) 1t_1
					(40,28) 1u
					(63,28) 1v
					(62,73) 1A^(2)
				  
				
				M^(2) 
 
 |e'| = 1    |e”| = k - 1
   σ 2(J_0, 1) = s
    c 2 = c(e_1) + c(e_2)k-1 + s 
			
			
			[t].3 
				[scale=1]/297.pdf  
					(51,28) 1t_1
					(40,28) 1u
					(63,28) 1v
					(39,73) 1A^(3)
				
				
				M^(3) 
 
 |e'| = k + 1
   σ 3(J_0, 1) = s
    c 3 = c(e_3)k+1 + s  
			
			figure 
		
		
		
		We verify Equation (<ref>):
		
    c^(1) + 2(k-1) c^(2) + 2(k+1) c^(3) = 
       2(kc(e_1) + c(e_2) + ks)+ (c(e_3) + ks) = 2 · 0 + 0 = 0.

		
		

		
		
		
  * Case {w2-r2-nd3-M-kk}: Assume that k_1 ≥ 2 and k_2 ≥ 2. Then k_3 > k_1, k_2, so Base I is precluded since |e_2|≠|e_3| nor |e_2|≠|e_4|. Base II.2.1.M, Base II.2.2.M, and Base II.1.M determine M 1, M 2 and M 3, respectively. See Figure <ref>.
		
		

		
		
		.5
			

			[scale=1.1]/256.pdf    
				(22,-6) 1t_2
				(75,-6) 1t_3
				(50,-6) 1w_0
				
				(-8,49) 1e_1	
				(-8,12) 1e_2	
				(102,23) 1e_3
				
				(50,67) 1A_0
			 
			

			
			M_0
		
		
		0.45
			
    c(e_1)k_1 + c(e_2)k_2 + s = 0
       c(e_3)k_1 + k_2 - 1 + s = 0

		
		
		

		
		  
			
			[t].3
				[scale=1]/259.pdf  
					(51,-6) 1t_1
					(32,-6) 1u
					(70,-6) 1v
					(62,52) 1A 1 
					
					(-10,49) 1e_1	
					(-10,13) 1e_2	
					(102,25) 1e_3
				
				
				

				
				M^(1) 
    |e'| = k_1 - 1    |e”| = k_2
       σ 1(J_0, 1) = s
       c^(1) = c(e_1)k_1-1 + c(e_2)k_2 + s
 
			
			
			[t].3 
				[scale=1]/260.pdf   
					(51,-6) 1t_1
					(32,-6) 1u
					(70,-6) 1v
					(65,66) 1A 2
				  
				
				

				
				M^(2)
				
    |e'| = k_1     |e”| = k_2 - 1
       σ 2(J_0, 1) = s
        c 2 = c(e_1)k_1 + c(e_2)k_2-1 +  s

			
			
			[t].3 
				[scale=1]/261.pdf 
					(51,-6) 1t_1
					(32,-6) 1u
					(70,-6) 1v
					(30,66) 1A 3
				
				
				

				
				M^(3)
				
    |e'| = k_1  + k_2
       σ 3(J_0, 1) = s
        c 3 = c(e_3)k_1 + k_2 +  s

			
			figure 
		
		
		
		We verify Equation (<ref>):
		
    (k_1 - 1) c^(1) + (k_2-1) c^(2) + (k_1 + k_2) c^(3) = 
        (k_1 + k_2 - 1k_1c(e_1) + k_1 + k_2 - 1k_2c(e_2) + (k_1 + k_2 - 1)s ) +  (c(e_3) + (k_1 + k_2 - 1) s  )= 0.

		
		

		
		
		
  * Case {w2-r2-nd3-P}: Assume that e_4 is above t_2, namely Cardinality P. Base I is precluded. Base II.2.1.P, Base II.2.2.P, Base II.1.P determine M 1, M 2 and M 3, respectively. See Figure <ref>.  

		
		
		.5
			[scale=1.3]/298.pdf  
				(27,-4) 1t_2
				(62,-4) 1t_3
				(43,-4) 1w_0
				(-1,30) 1e_1	
				(-1,8) 1e_2	
				(98,25) 1e_3	 
				(43,52) 1A_0
			 
			

			
			M_0
		
		0.45
			
    c(e_1)k_1  + c(e_2)k_2  + s = 0 
       c(e_3)k_1 + k_2 + 1  + s = 0

		
		
		
 
		
		
			
			[t].3
				[scale=1]/302.pdf  
					(44,-3) 1t_1
					(33,-3) 1u
					(55,-3) 1v
					(54,54) 1A^(1) 
					
					(-3,30) 1e_1	
					(-3,9) 1e_2	
					(98,25) 1e_3	
				

				
				M^(1) 
 

				|e'| = k_1 + 1    |e”| = k_2
   σ q(J_0, 1) = s
   c^(1) = c(e_1)k_1+1 + c(e_2)k_2 +  s
			
			
			[t].3 
				[scale=1]/303.pdf   
					(44,-3) 1t_1
					(33,-3) 1u
					(55,-3) 1v
					(32,54) 1A^(2)
				  
				
				

				
				M^(2) 
 

				|e'| = k_1     |e”| = k_2 + 1
   σ 2(J_0, 1) = s
   c^(2) = c(e_1)k_1  + c(e_2)k_2+1 +  s 
			
			
			[t].3 
				[scale=1]/304.pdf 
					(44,-3) 1t_1
					(33,-3) 1u
					(55,-3) 1v
					(54,54) 1A^(3)
				
				
				

				
				M^(3) 
 

				|e'| = k_1 + k_2
   σ 3(J_0, 1) = s
   c^(3) = c(e_3)k_1 + k_2 + s 
			 
			figure 
		
		
		
		We verify Equation (<ref>):
		
    (k_1 + 1) c^(1) + (k_2+1) c^(2) + (k_1 + k_2) c^(3) =
        (k_1 + k_2 + 1)( 1 k_1 c(e_1) +  1 k_2 c(e_2) + s) + (k_1 + k_2 + 1)( 1 k_1 + k_2 + 1 c(e_3) + s) = 0 + 0 = 0.

		
		

		
		
		
  * Case {w2-r2-nd2}: Assume that A_0 is 2. Then both e_1, e_4 are dangling. Let h be equal to h(e_2) equal to h(e_3).  
		
		
		
		
  * Case {w2-r2-nd2-M}: Recall that e_1 is above t_2. Assume that e_4 is above t_3. Then k_2 + 1 = |A_0| = k_3 + 1. Since |e_2| = |e_3|, the columns corresponding to t_2 and t_3 in A_0 are equal. So M_0 is not full-rank, contradicting Lemma <ref>. 
		
		
		
		
  * Case {w2-r2-nd2-P}: Assume that e_4 is above t_2. Then k_3 = |A_0| = k_2+2, and Base I is precluded. In Base II all vertices of  belong to Case (r1-nd2). Thus,  has a single edge |e'|. The Case (r1-nd2) on the end of e_3 above v implies that |e'| = k_3 - 1 = k_2 + 1. This is the only possibility, shown in Figure <ref>.
		
		
			[t].45
				[scale=1]/305.pdf 
					(43,41) 1A_0
					(28,-4) 1t_2
					(61,-4) 1t_3
					(43,-4) 1w_0	
					(-3,7) 1e_2	
					(97,18) 1e_3	 
				
				
				

				
				M_0 
 

				s = σ_0(J_0, 2) = σ_0(J_0, 3)
   c_hk_2 + s = 0,   c_hk_2+2 + s = 0
				
			
			
			[t].45 
				[scale=1]/306.pdf  
					(45,-4) 1t_1
					(32,-4) 1u 
					(56,-4) 1v 
				 
				
				

				
				M^(1) 
 

				|e'| = k_2 + 1
   σ 1(J_0, 1) = s
    c 1 = c_hk_2+1 + s
				
			
			figure 
		
		

		
		
		From c_h + k_2s = 0 and c_h + (k_2+2)s = 0 it follows that c 1 = c_h + (k_2+1)s = 0, hence M 1 is not full-dimensional.   
		
		
		
  * Case {w2-r1}: Assume that r_0(A_0) = 1. Then A_0 = 3. Since A_0 satisfies no-return, let e_2, e_3 be non-dangling edges in A_0, above t_2, t_3, respectively. Let e_1 be the remaining edge of A_0. We may assume without loss of generality that e_1 is above t_2. By the refinement property k_1 + k_2 = |A_0| = k_3. Let Ã be the unique vertex of  with r q(Ã) = 1. If Ã is above u (resp. v), then the vertices of  above v (resp. u) belong to the Case (r0-nd2) of the local properties (Case (r0-nd3) would contradict that v (resp. u) is divalent). These facts and Lemma <ref> determine the classes. 
		
		
  * Case {w2-r1-nd3}: Assume that A_0 is 3. We get two elements in M_0, shown in Figure <ref>.
		

		
		
		
			[t].30
				[scale=1]/307.pdf  
					(27,-5) 1t_2
					(62,-5) 1t_3
					(42,-5) 1w_0
					(-4,31) 1e_1	
					(-4,7) 1e_2	
					(96,20) 1e_3	 
					(43,42) 1A_0
				
				
				

				
				local part around A_0
				
    σ_0(J_A_0,2) = c(e_1)/k_1  + c(e_2)/k_2
       σ_0(J_A_0,3) = c(e_3)/k_1 + k_2

			
			
			[t].30 
				[scale=1]/308.pdf 
					(43,-4) 1t_1
					(32,-4) 1u
					(55,-4) 1v
					(31,42) 1A^(q)
				
				
				
				

				
				1 (Ã) = u
				
    |e'| = k_3 
       σ 1(J_A_0,1) = c(e_3)/k_1 + k_2

			
			
			[t].30 
				[scale=1]/309.pdf 
					(43,-4) 1t_1
					(32,-4) 1u
					(55,-4) 1v
					(54,42) 1A^(q)
				  
				
				

				
				2 (Ã) = v
				
    |e'| = k_1     |e”| = k_2
       σ 2(J_A_0,1) = c(e_1)k_1 + c(e_2)k_2

			
			figure 
		
		
		
 
		
		Thus, 
    σ 1(J_A_0,1) + σ 2(J_A_0,1) =  σ_0(J_A_0, 2) + σ_0(J_A_0, 3).

		
		

		
		
  * Case {w2-r1-nd2}: Assume that A_0 is 2. We may assume that e_1 is dangling. So k_3 = k_2 + 1. Let h = h(e_2) = h(e_3).  We get two elements in M_0, shown in Figure <ref>.
		
		

		
		
		
			[t].30
				[scale=1]/310.pdf  
					(25,-5) 1t_2
					(63,-5) 1t_3
					(42,-5) 1w_0
					(-1,7) 1e_2	
					(97,13) 1e_3	 
					(43,30) 1A_0
				
				
				

				
				local part around A_0
				
    σ_0(J_A_0,2) = c_hk_1
       σ_0(J_A_0,3) = c_hk_1 + 1

			
			
			[t].3 
				[scale=1]/311.pdf 
					(43,-4) 1t_1
					(32,-4) 1u
					(55,-4) 1v
					(31,30) 1A^(q)
				
				
				

				
				1 (Ã) = u
				
    |e'| = k_2 + 1
       σ 1(J_A_0,1)= c_hk_2 + 1
 
			 
			
			[t].3 
				[scale=1]/312.pdf  
					(43,-4) 1t_1
					(32,-4) 1u
					(55,-4) 1v
					(54,30) 1A^(q)
				  
				
				

				
				2 (Ã) = v
				
    |e'| = k_2
       σ 2(J_A_0,1) = c_hk_2

			
			figure 
		
		
		
 
		
		Thus, 
    σ 1(J_A_0,1) + σ 2(J_A_0,1) = σ_0(J_A_0, 2) + σ_0(J_A_0, 3).

		
		

		
		
  * Proof of (<ref>) for case {w2-r1}: There is another vertex B_0 above w_0 with r_0(B_0) = 1. The previous analysis holds for B_0, with notation entirely analogous. Note that in M q, q (Ã)  q (B̃) because u =  v = 1. So q (Ã) determines the gluing datum, and it still holds that 
    σ 1(J_B_0,1) + σ 2(J_B_0,1) =  σ_0(J_B_0, 2) + σ_0(J_B_0, 3).
 This gives the following calculation, which verifies Equation (<ref>): 
		
    c1    = σ 1(J_A_0,1) + σ 1(J_B_0,1) + s,
    		   
    		c^(2)   = σ 2(J_A_0,1) + σ 2(J_B_0,1) + s,

		
    c^(1) + c^(2)   = σ_0(2) + σ_0(3) = 0 + 0 = 0.

	 
	
	

 §.§ Conclusion
 It is remarkable how diverse the arguments of Section <ref> are. By no means do we stand in front of a construction that has been repeated with subtle variations. The richness and diversity of the behaviour of possibly full-dimensional gluing datums defied many attempts of further consolidation into fewer cases. The end result is exhaustive, so Lemma <ref> is verified. This finishes the proof of Theorem <ref>. The method is effective as well; see <cit.> for code.
	
	

 §.§ Preview part II
 To summarize the past sneak peeks, let 1, 2 be
	that share a limit _0. The cone C__0
	may be viewed as a common face to the cones C_ 1
	and C_ 2. Hence, it is natural to glue at
	C__0 and mod out by isomorphisms.  The possibilities
	for C__0 include the cases that we skipped where
	the source of _0 is not trivalent. The moduli space d g 
	that emerges by gluing all genus-g full-dimensional cones, as mentioned in Subsection <ref>, is the main object in Part II of
	this series. We study the connectivity of this space and its projection to g to get a
	stronger version of Theorem <ref>, in the sense that we obtain a count of realizations. Also, the result becomes independent from the work in cd18.
	
	