Packing and covering induced subdivisions

Abstract

A class F of graphs has the induced Erdos-Posa 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 F, or a vertex set of size at most f(k) hitting all induced copies of graphs in 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 Erdos-Posa property if and only if \ell \leq 4. In this paper, we investigate whether or not the class of H-subdivisions has the induced Erdos-Posa 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 Erdos-Posa 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 Erdos-Posa 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 Erdos-Posa property.