Chordal Graphs, Even-Hole-Free Graphs and Sparse Obstructions to Bounded Treewidth

Even-hole-free graphs pose a central challenge in identifying hereditary classes of bounded treewidth. We investigate this matter by presenting and studying the following conjecture: for an integer $t\ge 4$ and a graph $H$, every even-hole-free graph of large enough treewidth has an induced subgraph isomorphic to either $K_t$ or $H$, if (and only if) $H$ is a $K_4$-free chordal graph. The “only if” part follows from the properties of the so-called layered wheels, a construction by Sintiari and Trotignon consisting of (even-hole, $K_4$)-free graphs with arbitrarily large treewidth. Alecu et al. proved recently that the conjecture holds in two special cases: (a) when $t=4$; and (b) when $H=\text{cone}\left(F\right)$ for some forest $F$; that is, $H$ is obtained from a forest $F$ by adding a universal vertex. Our first result is a common strengthening of (a) and (b): for an integer $t\ge 4$ and graphs $F$ and $H$, (even-hole, $\text{cone}\left(\text{cone}\left(F\right)\right),H,K_t$)-free graphs have bounded treewidth if and only if $F$ is a forest and $H$ is a $K_4$-free chordal graph. Also, for general $t\ge 4$, we push the current state of the art further than (b) by settling the conjecture for the smallest choices of $H$ that are not coned forests. The latter follows from our second result: we prove the conjecture when $H$ is a crystal; that is, a graph obtained from arbitrarily many coned double stars by gluing them together along the “middle” edges of the double stars. In the first version of this paper, we suggested the following which is a strengthening of our main conjecture: for every $t\ge 1$, every graph of sufficiently large treewidth has an induced subgraph of treewidth $t$ which is either complete, complete bipartite, or 2-degenerate. This strengthening has now been refuted by Chudnovsky and Trotignon [On treewidth and maximum cliques, arxiv:2405.07471, 2024].