Induced Forest Minor Theorem for Graphs Without an Induced Star

2026-09-09

AI Summary

研究通过路径分解方法解决了排除特定诱导子图的图类中的路径独立数问题，证明了此类图的路径独立数有界，并提供多项式时间算法解决相关NP难问题。

Abstract

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.

Problem

• path independence number
• induced forest minor
• induced star
• tree independence number
• polynomial-time algorithm

Innovation

• path independence number
• induced forest minor
• induced star
• tree independence number
• polynomial-time algorithm