Given a collection of n rooted trees with depth h, we give a necessary and sufficient condition for this collection to be the collection of h-depth universal covering neighborhoods at each vertex.
@cite_1: 
@cite_7: Seven criteria for integer sequences being graphic are listed. Being graphic means that there is a simple graph with the given integer sequence as degree sequence. One of the criteria leads to a new and constructive proof of the well-known criterion of Erdos-Gallai.
@cite_8: Abstract Given pairs ( a 1 , b 1 ) , … , ( a n , b n ) of nonnegative integers, the digraph realization problem for digraphs asks whether there is a simple digraph (no loops or multiple arcs) with vertices v 1 , … , v n such that each vertex v i ∈ V has indegree a i and outdegree b i . Fulkerson and Chen obtained a characterization analogous to the classical Erdős–Gallai characterization for graphs, but with the additional constraint that the pairs must be sorted in nonincreasing lexicographical order. We provide a more general characterization that avoids the additional sorting. The inequalities needed correspond to those k such that a k + 1 > a k . We prove a similar result when one loop is allowed at each vertex.
A short and constructive proof is available at @cite_1 . In fact, the Erdös-Gallai condition is not the only sufficient and necessary condition for an integer sequence to be graphical ; there are some other (equivalent) conditions, notably listed in @cite_7 . The corresponding realization problem for had also been solved quite early ; see the interesting note @cite_8 for a complete history and presentation of the many variants.