无诱导星图的诱导森林子式定理
Induced Forest Minor Theorem for Graphs Without an Induced Star
- Université libre de Bruxelles（布鲁塞尔自由大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究图的路径独立数，证明排除诱导森林子式和诱导星图的图类具有有界路径独立数，部分解决猜想，并推广至树独立数，从而为相关图类上的最大权独立集等问题提供多项式时间算法。
AI中文摘要：
受近期关于树独立数研究工作的启发，我们研究了图 $G$ 的路径独立数：即最小的整数 $k$，使得存在 $G$ 的一个路径分解，其中每个袋子诱导出一个独立数至多为 $k$ 的图。我们证明了每个同时排除诱导森林子式和诱导星图的图都具有有界的路径独立数。这刻画了排除诱导星图的图类何时具有有界路径独立数，同时也部分解决了 Dallard、Krnc、Kwon、Milanič、Munaro、Štorgel 和 Wiederrecht（2024 年）的一个猜想。此外，我们证明了同时排除 apex-森林诱导子式和诱导星图的图具有有界的树独立数。作为推论，对于每个固定的 apex-森林 $H$ 和整数 $t$，存在一个多项式时间算法来测试一个 $K_{1,t}$-诱导子图-free 图是否包含 $H$ 作为诱导子式。而且，由此可知，最大权独立集问题以及其他几个 NP-困难问题可以在排除 $H$ 作为诱导子式的 $K_{1,t}$-诱导子图-free 图上于多项式时间内求解。
英文摘要：
Motivated by recent work on tree independence number, we study the path independence number of a graph $G$: the minimum integer $k$ such that there is a path decomposition of $G$ where each bag induces a graph with independence number at most $k$. We show that every graph excluding both an induced forest minor and an induced star has bounded path independence number. This characterises when a graph class that excludes an induced star has bounded path independence number while also partially resolving a conjecture of Dallard, Krnc, Kwon, Milani{č}, Munaro, Štorgel and Wiederrecht (2024). Furthermore, we show that graphs excluding both an apex-forest induced minor and an induced star have bounded tree independence number. As a consequence, for every fixed apex-forest $H$ and integer $t$, there is a polynomial-time algorithm to test whether a $K_{1,t}$-induced-subgraph-free graph contains $H$ as an induced minor. Moreover, it follows that the Maximum Weight Independent Set problem, as well as several other NP-hard problems, can be solved in polynomial-time on $K_{1,t}$-induced-subgraph-free graphs that exclude $H$ as an induced minor.
