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

中文翻译：

---

(Theta,triangle)-free 和 (even hole, $K_4$)-free 图。第 2 部分：树宽的边界

{\em theta} 是由三个内部顶点不相交的无弦路径组成的图 $P_1 = a \dots b$, $P_2 = a \dots b$, $P_3 = a \dots b$，长度至少为~2并且除了与 $a$ 相关的三个边和与 $b$ 相关的三个边之外，路径之间不存在任何边。{\em 金字塔} 是由三个无弦路径组成的图 $P_1 = a \dots b_1$, $P_2 = a \dots b_2$, $P_3 = a \dots b_3$，长度至少为~1，其中两条长度至少为 2，顶点不相交，除了在 $a$ 处，并且 $b_1b_2b_3$ 是一个三角形，除了三角形的路径和与 ~$a$ 相关的三个边之外，路径之间不存在边。\emph{偶数孔} 是偶数长度的无弦循环。对于三个非负整数 $i\leq j\leq k$，令 $S_{i,j,k}$ 为顶点为 $v$ 的树，从它开始三个路径为 $i$, $j$ , 和 $k$ 边分别。我们用 $K_t$ 表示 $t$ 顶点上的完整图。我们证明对于所有非负整数 $i, j, k$，不包含 theta、不包含 $K_3$ 和不包含 $S_{i, j, k}$ 作为诱导子图的图类具有有界树宽。我们证明对于所有非负整数 $i, j, k, t$，不包含偶数孔、不包含金字塔、不包含 $K_t$ 和不包含 $S_{i, j, k}$ 的图类为诱导子图具有有界树宽。为了限制树宽，我们证明每个大树宽的图必须包含一个大集团或大基数的最小分隔符。不包含偶数孔、金字塔、$K_t$ 和 $S_{i, j, k}$ 作为诱导子图的图类具有有界树宽。为了限制树宽，我们证明每个大树宽的图必须包含一个大集团或大基数的最小分隔符。不包含偶数孔、金字塔、$K_t$ 和 $S_{i, j, k}$ 作为诱导子图的图类具有有界树宽。为了限制树宽，我们证明每个大树宽的图必须包含一个大集团或大基数的最小分隔符。