Packing and covering induced subdivisions
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A graph class F has the induced Erdős-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 F, or a vertex set of size at most f(k) hitting all induced copies of graphs in F. Kim and Kwon (SODA'18) showed that for a cycle C_ℓ of length ℓ, the set of C_ℓ-subdivisions has the induced Erdős-Pósa property if and only if ℓ< 4. In this paper, we investigate whether subdivisions of H have the induced Erdős-Pósa 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 Erdős-Pósa property. For this, we describe three basic constructions that can be used to prove that subdivisions of a graph do not have the induced Erdős-Pósa property. Among remaining graphs, we prove that the subdivisions of the diamond, the 1-pan, and the 2-pan have the induced Erdős-Pósa property.
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