On the tree-width of even-hole-free graphs

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019]. The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary. We prove that for every graph $G$, if $G$ excludes a fixed graph $H$ as a minor, then $G$ either has small tree-width, or $G$ contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymour's excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graph have bounded tree-width [da Silva and Linhares Sales, Discrete Applied Mathematics 2010]. We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the tree-width of the class of (even hole, pyramid)-free graphs of degree at most 4.