诱导打包树宽 II. 排除团或双团

中文摘要 AI 辅助

诱导打包树宽的概念旨在将通过禁止诱导子图或诱导 minors 定义的类与通过存在某种结构化树分解定义的类统一起来。对于图 $H$，\\(\treepi_{H}\\) 表示的 \\(\emph{诱导 $H$-打包树宽}\\) 是一种基于树分解的图参数，对于每个袋子，它衡量与该袋子相交的成对反完备的 $H$ 诱导副本的最大数量。这一概念推广了先前研究的一些参数：当 $H=P_1$ 时，它等价于树独立数；当 $H=P_2$ 时，它等价于诱导匹配树宽。我们证明了以下结果：\begin{itemize}[itemsep=2mm,leftmargin=6mm] \item 对于所有 $a,t\in \mathbb{N}$，具有有界诱导 $P_t$-打包树宽的 $K_{a,a}$-自由图具有有界树独立数。这扩展了 Abrishami 等人 [SIAM J. Discrete Math., 2025] 关于 $t=2$ 的先前结果，以及 Hajebi 和 Spirkl 的结果，他们证明了 $(P_t,K_{a,a})$-自由图具有有界树独立数。\item 如果 $H$ 是任何固定路径或星形图，那么具有有界诱导 $H$-打包树宽的图类是 $\chi$-有界的。同样，这扩展了 Abrishami 等人 [SIAM J. Discrete Math., 2025] 关于 $H=P_2$ 的先前结果。\item 最后，我们研究了诱导打包树宽与 \emph{sim-width}（一种基于分支分解的宽度参数）之间的关系。我们表明，尽管 \emph{sim-width} 和诱导 $P_3$-打包树宽不可比较，但具有有界 sim-width 且排除所有 \emph{$H$-障碍}（某些迫使大诱导 $H$-打包树宽的图）的图具有有界诱导 $H$-打包树宽。这同时推广并解决了 Abrishami 等人 [SIAM J. Discrete Math., 2025] 和 Brettell 等人 [European J. Comb., 2025] 提出的问题。\end{itemize}

英文摘要

The notion of induced packing treewidth aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. For a graph $H$, \emph{induced $H$-packing treewidth}, denoted by $\treepi_{H}$, is a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of $H$ intersecting that bag. This notion generalizes some previously studied parameters: when $H=P_1$, it is equivalent to tree-independence number, and when $H=P_2$, it is equivalent to induced matching treewidth. We prove the following: \begin{itemize}[itemsep=2mm,leftmargin=6mm] \item For all $a,t\in \mathbb{N}$, $K_{a,a}$-free graphs of bounded induced $P_t$-packing treewidth have bounded tree-independence number. This extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $t=2$, and a result of Hajebi and Spirkl who showed that $(P_t,K_{a,a})$-free graphs have bounded tree-independence number. \item If $H$ is any fixed path or a star, then the class of graphs of bounded induced $H$-packing treewidth is $χ$-bounded. Again, this extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $H=P_2$. \item Finally, we study the relationship between induced packing treewidth and \emph{sim-width}, a width parameter based on branch decompositions. We show that, although \emph{sim-width} and induced $P_3$-packing treewidth are incomparable, graphs of bounded sim-width that exclude all \emph{$H$-obstructions}---certain graphs that force large induced $H$-packing treewidth---have bounded induced $H$-packing treewidth. This simultaneously generalizes and resolves questions posed by Abrishami et al. [SIAM J. Discrete Math., 2025] and Brettell et al. [European J. Comb., 2025]. \end{itemize}

发表机构

• IT University of Copenhagen（哥本哈根信息技术大学）
• Warsaw University of Technology（华沙理工大学）

机构由 AI 辅助整理，请以论文原文为准。