(Theta, triangle)‐free and (even hole, K4)‐free graphs. Part 2: Bounds on treewidth

A {\em theta} is a graph made of three internally vertex-disjoint chordless paths $P_1 = \dots b$, $P_2 $P_3 b$ length at least~2 and such that no edges exist between the except incident to $a$ $b$. pyramid} b_1$, b_2$, b_3$ least~1, two which have least 2, $a$, $b_1b_2b_3$ triangle those to~$a$. An \emph{even hole} cycle even length. For non-negative integers $i\leq j\leq k$, let $S_{i,j,k}$ be tree with vertex $v$, from start $i$, $j$, $k$ respectively. We denote by $K_t$ complete on $t$ vertices. prove for all $i, j, class graphs contain theta, $K_3$, $S_{i, k}$ as induced subgraphs bounded treewidth. k, t$, hole, pyramid, $K_t$, To bound treewidth, we every large treewidth must clique or minimal separator cardinality.