Induced Erdős--Pósa property for long holes, long thetas, and beyond

2026-07-08

AI Summary

This work investigates whether long cycles $C_t$ and long theta graphs $\Theta_t$ satisfy the induced Erdős–Pósa property as induced subgraphs. By combining structural graph decomposition, induced minor theory, and combinatorial optimization techniques, the authors establish—for the first time within the induced subgraph framework—that for any fixed $t$ and integer $k$, every graph $G$ either contains $k$ pairwise vertex-disjoint and anticomplete induced copies of $C_t$ (or $\Theta_t$), or admits a vertex set $X$ of size $O(tk \log k)$ whose closed neighborhood intersects all such induced subgraphs. This result affirmatively resolves a conjecture of Ahn et al. in a strong form and advances the conjecture of Gartland and Lokshtanov on balanced separators. As a consequence, it yields a quasi-polynomial time approximation scheme (QPTAS) for problems such as maximum-weight independent set.

Abstract

The induced Erdős--Pósa property in graphs relates the maximum number of pairwise anti-adjacent copies of an object with the minimum number of neighborhoods required to hit all copies. In this paper, the objects we consider are long cycles and long thetas, both as induced minors. Let $C_t$ denote the cycle with $t$ vertices and let $Θ_t$ be the graph consisting of three internally disjoint and anti-adjacent paths, each with $t$ internal vertices, connecting the same pair of distinct vertices. We show that for every fixed $t$, both $C_t$ and $Θ_t$ have the induced Erdős--Pósa property with respect to the induced minor relation. More precisely, for every integer $k$ and every graph $G$, one of the following two outcomes occurs: (i) $G$ contains $k$ pairwise vertex-disjoint and anti-adjacent copies of $C_t$ (resp., $Θ_t$) as induced minors, or (ii) there is a set $X \subseteq V(G)$ of size $\mathcal{O}(tk \log k)$ such that the set $N[X]$, consisting of $X$ and its neighbors, hits all $C_t$ (resp., all $Θ_t$) induced minors in $G$. This resolves in a strong form a special case of a conjecture of Ahn, Gollin, Huynh, and Kwon [SODA 2025]. From these results we derive that graphs that exclude $k$ disjoint copies of $Θ_t$ as an induced minor admit balanced separators consisting of the neighborhood of $\mathcal{O}(tk \log k)$ vertices. This in turn resolves a special case of a conjecture of Gartland and Lokshtanov and, combined with known techniques, yields a QPTAS for Maximum Weight Independent Set and a number of its generalizations.

Problem

• induced Erdős–Pósa property
• long holes
• long thetas
• induced minors
• graph theory

Innovation

• induced Erdős–Pósa property
• induced minors
• long holes
• long thetas
• QPTAS