无（theta，三角形）图与无（偶洞，$K_4$）图。第2部分：树宽的界
(Theta, triangle)-free and (even hole, $K_4$)-free graphs. Part 2 : bounds on treewidth
AI总结：
本文证明了无（theta，三角形）图与无（偶洞，pyramid，$K_t$）图在排除特定导出子图后具有有界树宽，并通过证明大树宽图必含大团或大基数极小分离集来界定树宽。
摘要：
{\em theta} 是由三条长度至少为 2、内部顶点互不相交的无弦路径 $P_1 = a \dots b$、$P_2 = a \dots b$、$P_3 = a \dots b$ 构成的图，且除了与 $a$ 关联的三条边和与 $b$ 关联的三条边外，这些路径之间不存在其他边。{\em pyramid} 是由三条长度至少为 1 的无弦路径 $P_1 = a \dots b_1$、$P_2 = a \dots b_2$、$P_3 = a \dots b_3$ 构成的图，其中两条路径长度至少为 2，它们除顶点 $a$ 外互不相交，且 $b_1b_2b_3$ 构成一个三角形，除了该三角形的边和与 $a$ 关联的三条边外，路径间无其他边。\emph{偶洞} 是长度为偶数的无弦环。对于三个非负整数 $i\leq j\leq k$，令 $S_{i,j,k}$ 为具有顶点 $v$ 的树，从 $v$ 出发分别有三条包含 $i$、$j$ 和 $k$ 条边的路径。我们用 $K_t$ 表示具有 $t$ 个顶点的完全图。我们证明，对于所有非负整数 $i, j, k$，不包含 theta、$K_3$ 和 $S_{i, j, k}$ 作为导出子图的图类具有有界树宽。我们证明，对于所有非负整数 $i, j, k, t$，不包含偶洞、pyramid、$K_t$ 和 $S_{i, j, k}$ 作为导出子图的图类具有有界树宽。为了限定树宽，我们证明每个具有大树宽的图必然包含大团或大基数极小分离集。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
