Abstract

Roughly, a graph has small “tree-width ” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: (i) a minimax formula relating tree-width with the largest such obstructions (ii) an association between such obstructions and large grid minors of the graph (iii) a “tree-decomposition ” of the graph into pieces corresponding with the obstructions. These results will be of use in later papers. 0 1991 Academic Press, Inc. 1. TANGLES Graphs in this paper are finite and undirected and may have loops or multiple edges. The vertex- and 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 * This work ’ was performed under a consulting agreement with Bellcore.

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