Forbidding anticomplete planar minors: Induced Erdős--Pósa property and Maximum Independent Set in QP

2026-07-10

AI Summary

This study investigates the structural and algorithmic properties of graphs excluding $k$ pairwise non-adjacent induced copies of a fixed planar graph $H$ (so-called $kH$-free graphs). It establishes, for the first time, an Erdős–Pósa property for planar induced subgraphs: there exists a function $f(k,H)$ such that removing the closed neighborhoods of at most $f(k,H)$ vertices suffices to eliminate all induced $H$-subgraphs. Leveraging this result, the authors prove that sparse $kH$-free graphs have treewidth $O(\log n)$. Combining protrusion decomposition with sparsity techniques, they design a quasipolynomial-time algorithm for the Maximum Independent Set problem on such graphs. This work extends classical minor theory to the induced setting and reveals a profound connection between local deletion operations and global sparsity structure.

Abstract

The Erdős--Pósa theorem asserts that every graph $G$ with no $k$ disjoint cycles contains a set $X$ of $f(k)$ vertices such that $G\setminus X$ has no cycle. Robertson and Seymour showed that this Erdős--Pósa property also holds for $H$-minor models of any planar graph $H$. Equivalently, if $G$ has no $k$ minor models of $H$ pairwise at distance at least 1 (i.e. disjoint), then one can remove $f(k,H)$ balls of radius 0 (i.e. vertices) to make the graph $H$-minor free. We show that this coarse graph theory point of view generalizes to distance at least 2 versus radius 1 balls, yielding the induced Erdős--Pósa property for planar minors. Namely, every graph $G$ which does not contain $k$ pairwise non-adjacent minor models of a planar graph $H$ (we say that $G$ is $kH$-free) can be made $H$-minor free by removing $f(k,H)$ neighborhoods. The proof relies on the fact that sparse $kH$-free graphs have linearly many independent large protrusions. The same method gives that sparse $kH$-free graphs can be made $H$-minor free by deleting $O(\log n)$ vertices (and thus have logarithmic tree-width). This gives a quasi-polynomial algorithm for the Maximum Independent Set problem for $kH$-free graphs.

Problem

• induced Erdős–Pósa property
• planar minors
• Maximum Independent Set
• kH-free graphs
• quasi-polynomial algorithm

Innovation

• induced Erdős–Pósa property
• planar minors
• kH-free graphs
• maximum independent set
• quasi-polynomial algorithm