Induced subgraphs and tree decompositions IV. (Even hole, diamond, pyramid)-free graphs

Tara Abrishami, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

Published 2022-03-13Version 1

A hole in a graph $G$ is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph $K_4$ by removing an edge. A pyramid is a graph consisting of a triangle called the base, a vertex called the apex, and three internally disjoint paths starting at the apex and disjoint otherwise, each joining the apex to a vertex of the base. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Cameron, da Silva, Huang, and Vu\v{s}kovi\'c proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, $K_4$)-free graphs of arbitrarily large treewidth, and observing induced diamonds all over their construction, they conjectured that (even hole, diamond, $K_4$)-free graphs have bounded treewidth. We approach this conjecture by showing that for every $t$, (even hole, pyramid, diamond, $K_t$)-free graphs have bounded treewidth. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses "non-crossing decompositions" methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of "non-crossing decompositions" to prove bounded treewidth in a graph class of unbounded maximum degree.