(Theta, triangle)‐free and (even hole, K4)‐free graphs. Part 2: Bounds on treewidth

Abstract

AbstractA theta is a graph made of three internally vertex‐disjoint chordless paths , of length at least 2 and such that no edges exist between the paths except the three edges incident to and the three edges incident to . A pyramid is a graph made of three chordless paths , of length at least 1, two of which have length at least 2, vertex‐disjoint except at , and such that is a triangle and no edges exist between the paths except those of the triangles and the three edges incident to . An even hole is a chordless cycle of even length. For three nonnegative integers , let be the tree with a vertex , from which start three paths with , and edges, respectively. We denote by the complete graph on vertices. We prove that for all nonnegative integers , the class of graphs that contain no theta, no , and no as induced subgraphs have bounded treewidth. We prove that for all nonnegative integers , the class of graphs that contain no even hole, no pyramid, no , and no as induced subgraphs have bounded treewidth. To bound the treewidth, we prove that every graph of large treewidth must contain a large clique or a minimal separator of large cardinality.