Graph minors. I. Excluding a forest

The path-width of a graph is the minimum value ofk such that the graph can be obtained from a sequence of graphsG₁,…,G_r each of which has at mostk + 1 vertices, by identifying some vertices ofG_i pairwise with some ofG_i+1 (1 ≤ i < r). For every forestH it is proved that there is a numberk such that every graph with no minor isomorphic toH 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 ifP is any property of graphs such that some forest does not have propertyP, then the set of minor-minimal graphs without propertyP is finite.