Graph minors. X. Obstructions to tree-decomposition

Graphs in this paper are finite and undirected and may have loops or multiple edges. The vertexand edge-sets of a graph G are denoted by V(G) and E(G). If G, = ( V1, E,), G2 = ( V2, E2) are subgraphs of a graph G, we denote the graphs (V1n V2,E1nE,) and (V,u V2, EluEZ) by G,nG, and G, u GZ, respectively. A separation of a graph G is a pair (G,, G2) of subgraphs with G1 u G2 = G and E(G1 n G2) = 0, and the order of this separation is f V(G, n G2)(. It sometimes happens with a graph G that for each separation (G, , G2) of G of low order, we may view one of G1, G, as the “main part” of G, in