the -minor-free-graphs have unbounded pathwidth. In the other direction, if contains an -vertex forest, then the -minor-free graphs have pathwidth at most . Obstructions to bounded pathwidth The property of having pathwidth at most is, itself, closed under taking minors: if has a path-decomposition with width at most , then the same path-decomposition remains valid if any edge is removed from , and any vertex can be removed from and from its path-decomposition without increasing the width. Contraction of an edge, also, can be accomplished without increasing the width of the decomposition, by merging the sub-paths representing the two endpoints of the contracted edge. Therefore, the graphs of pathwidth at most can be characterized by a set of excluded minors. Although necessarily includes at least one forest, it is not true that all graphs in are forests: for instance, consists of two graphs, a seven-vertex tree and the triangle . However, the set of trees in may be precisely characterized: these trees are exactly the trees that can be formed from three trees in by connecting a new root vertex by an edge to an arbitrarily chosen vertex in each of the three smaller trees. For instance, the seven-vertex tree in is formed in this way from the two-vertex tree (a single edge) in . Based on this construction, the number of forbidden minors in can be shown to be at least . The complete set of forbidden minors for pathwidth-2 graphs has been computed; it contains 110 different graphs. Structure theory The graph structure theorem for minor-closed graph families states that, for any such family , the graphs in can be decomposed into clique-sums of graphs that can be embedded onto surfaces of bounded genus, together with a bounded number of apexes and vortices for each component of the clique-sum. An apex is a vertex that may be adjacent to any other vertex in its component, while a vortex is a graph of bounded pathwidth that is glued into one of the faces of the bounded-genus embedding of a component. The cyclic ordering of the vertices around the face into which a vortex is embedded must be compatible with the path decomposition of the vortex, in the sense that breaking the cycle to form a linear ordering must lead to an ordering with bounded vertex separation number. This theory, in which pathwidth is intimately connected to arbitrary minor-closed graph