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Piercing independent sets in graphs without large induced matching

Jiangdong Ai, Hong Liu, Zixiang Xu, Qiang Zhou

Abstract

Given a graph $G$, denote by $h(G)$ the smallest size of a subset of $V(G)$ which intersects every maximum independent set of $G$. We prove that any graph $G$ without induced matching of size $t$ satisfies $h(G)\le ω(G)^{3t-3+o(1)}$. This resolves a conjecture of Hajebi, Li and Spirkl (Hitting all maximum stable sets in $P_{5}$-free graphs, JCTB 2024).

Piercing independent sets in graphs without large induced matching

Abstract

Given a graph , denote by the smallest size of a subset of which intersects every maximum independent set of . We prove that any graph without induced matching of size satisfies . This resolves a conjecture of Hajebi, Li and Spirkl (Hitting all maximum stable sets in -free graphs, JCTB 2024).
Paper Structure (2 sections, 2 theorems, 2 equations)

This paper contains 2 sections, 2 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. The proof

Key Result

Theorem 1.2

Let $G$ be a graph without induced matching of size $t$, then $h(G)\le 10t^t\omega(G)^{3t-3}\log\omega(G)$.

Theorems & Definitions (3)

  • Conjecture 1.1: 2023P5Free
  • Theorem 1.2
  • Theorem 2.1: 1987DCG