Packing and covering induced subdivisions

TL;DR

This paper investigates which graphs have the induced Erdős-Pósa property for their subdivisions, fully characterizing forests and bipartite graphs, and identifying necessary conditions for other graphs.

Contribution

It completely characterizes when the class of H-subdivisions has the induced Erdős-Pósa property for forests and bipartite graphs, and provides necessary conditions for other graphs.

Findings

Forests and complete bipartite graphs have the induced Erdős-Pósa property for their subdivisions.

The class of H-subdivisions has the induced Erdős-Pósa property for diamonds, 1-pan, and 2-pan graphs.

Necessary conditions are identified for graphs H to have the property, with constructions showing when it does not hold.

Abstract

A class $\mathcal{F}$ of graphs has the induced Erd\H{o}s-P\'osa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA'18) showed that for a cycle $C_{\ell}$ of length $\ell$, the class of $C_{\ell}$-subdivisions has the induced Erd\H{o}s-P\'osa property if and only if $\ell\le 4$. In this paper, we investigate whether or not the class of $H$-subdivisions has the induced Erd\H{o}s-P\'osa property for other graphs $H$. We completely settle the case when $H$ is a forest or a complete bipartite graph. Regarding the general case, we identify necessary conditions on $H$ for the class of $H$-subdivisions to have the induced Erd\H{o}s-P\'osa property. For this, we provide three basic constructions that are useful to prove that the class of the subdivisions of a graph does not have the induced Erd\H{o}s-P\'osa property. Among remaining graphs, we prove that if $H$ is either the diamond, the $1$-pan, or the $2$-pan, then the class of $H$-subdivisions has the induced Erd\H{o}s-P\'osa property.