A theta is a graph made of three internally vertex-disjoint chordless paths $P_1=a\text{…}b,P_2=a\text{…}b$, $P_3=a\text{…}b$ 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 pyramid is a graph made of three chordless paths $P_1=a\text{…}{b}_1,P_2=a\text{…}{b}_2$, $P_3=a\text{…}{b}_3$ of length at least 1, two of which have length at least 2, vertex-disjoint except at $a$, and such that $b_1{b}_2{b}_3$ is a triangle and no edges exist between the paths except those of the triangles and the three edges incident to $a$. An even hole is a chordless cycle of even length. For three nonnegative integers $i\le j\le k$, 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 nonnegative 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 nonnegative 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：

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(Theta,triangle)-free 和 (even hole, K 4)-free 图。第 2 部分：树宽的界限

甲THETA是曲线图由三个内部顶点不相交chordless路径${磷}_1=\mathrm{一种}\text{…}乙,{磷}_2=\mathrm{一种}\text{…}乙$, ${磷}_3=\mathrm{一种}\text{…}乙$ 长度至少为 2 并且路径之间不存在边，除了三个与 $\mathrm{一种}$ 和三个边缘事件 $乙$. 甲金字塔是曲线图由三个chordless路径${磷}_1=\mathrm{一种}\text{…}{乙}_1,{磷}_2=\mathrm{一种}\text{…}{乙}_2$, ${磷}_3=\mathrm{一种}\text{…}{乙}_3$ 长度至少为 1，其中两个长度至少为 2，顶点不相交，除了在 $\mathrm{一种}$，并且这样 ${乙}_1{乙}_2{乙}_3$ 是一个三角形，路径之间不存在边，除了三角形和三个边与 $\mathrm{一种}$. 一个连孔是偶数长度的无线周期。对于三个非负整数$\mathrm{一世}\le j\le 克$， 让 ${秒}_{\mathrm{一世},j,克}$ 成为有顶点的树 $v$，从三个路径开始 $\mathrm{一世},j$， 和 $克$边，分别。我们表示为${钾}_{吨}$ 上的完整图 $吨$顶点。我们证明对于所有非负整数$\mathrm{一世},j,克$，不包含 theta 的图类，不包含 ${钾}_3$， 和不 ${秒}_{\mathrm{一世},j,克}$因为诱导子图具有有界树宽。我们证明对于所有非负整数$\mathrm{一世},j,克,吨$，不包含偶数孔，不包含金字塔，不包含 ${钾}_{吨}$， 和不 ${秒}_{\mathrm{一世},j,克}$因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。