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arXiv · 2609.10406

Induced Forest Minor Theorem for Graphs Without an Induced Star

Robert Hickingbotham·Gwenaël Joret
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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{\v{c}}, Munaro, \v{S}torgel 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.

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BibTeXRIS

Robert Hickingbotham, Gwenaël Joret. 2026-09-09. Induced Forest Minor Theorem for Graphs Without an Induced Star. https://arxiv.org/abs/2609.10406

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