Packing and covering induced subdivisions

A graph class $\mathcal{F}$ 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 set 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 subdivisions of $H$ have the induced Erd\H{o}s-P\'osa property for other graphs $H$. We completely settle the case where $H$ is a forest or a complete bipartite graph. Regarding the general case, we identify necessary conditions on $H$ for its subdivisions to have the induced Erd\H{o}s-P\'osa property. For this, we describe three basic constructions that can be used to prove that subdivisions of a graph do not have the induced Erd\H{o}s-P\'osa property. Among remaining graphs, we prove that the subdivisions of the diamond, the 1-pan, and the 2-pan have the induced Erd\H{o}s-P\'osa property.