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

The class of all even-hole-free graphs has unbounded tree-width, as it contains complete graphs. Recently, a (even-hole, K4)-free was constructed, that still tree-width (Sintiari and Trotignon, 2019). degree arbitrarily large clique-minors. We ask whether this is necessary. prove for every graph G, if G excludes fixed H minor, then either small or wall the line induced subgraph. This can be seen strengthening Robertson Seymour’s excluded grid theorem case minor-free Our implies excluding minor bounded tree-width. In fact, our applies to more general class: (theta, prism)-free known result planar even hole-free have (Silva et al., 2010). conjecture If true, would mean even-hole-freeness testable in bounded-degree model property testing. subcubic we give bound on (even hole, pyramid)-free at most 4.