Graph minors. I. Excluding a forest

Abstract

The path-width of a graph is the minimum value of k such that the graph can be obtained from a sequence of graphs G 1,…,G r each of which has at most k + 1 vertices, by identifying some vertices of G i pairwise with some of G i+1 ( 1 ≤ i < r). For every forest H it is proved that there is a number k such that every graph with no minor isomorphic to H has path-width ≆k. This, together with results of other papers, yields a “good” algorithm to test for the presence of any fixed forest as a minor, and implies that if P is any property of graphs such that some forest does not have property P, then the set of minor-minimal graphs without property P is finite.