Packing and Covering Induced Subdivisions

Abstract

A class $\mathcal{F}$ of graphs has the induced Erdös--Pósa 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 in [J. Combin. Theory Ser. B, 145 (2020), pp. 65--112] showed that for a cycle $C_{\ell}$ of length $\ell$, the class of $C_{\ell}$-subdivisions has the induced Erdös--Pósa property if and only if $\ell\le 4$. In this paper, we investigate whether or not the class of $H$-subdivisions has the induced Erdös--Pósa 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 Erdös--Pósa property. For this, we provide three basic constructions that are useful for proving that the class of the subdivisions of a graph does not have the induced Erdös--Pósa property. Among remaining graphs, we prove that if $H$ is the diamond, the 1-pan, or the 2-pan, then the class of $H$-subdivisions has the induced Erdös--Pósa property.