  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.

Discrete Mathematics;Data Structures and Algorithms;Combinatorics