2018/03/20 by Kwon, O-joung, Raymond, Jean-Florent
#05C75 #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #G.2.2
paper · doi:10.48550/arxiv.1803.07581
A class F of graphs 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 Cℓ of length ℓ, the class of Cℓ-subdivisions has the induced Erdős-Pósa property if and only if ℓ≤ 4. In this paper, we investigate whether or not the class of H-subdivisions has the induced Erdős-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 Erdős-Pósa 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 Erdős-Pósa 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 Erdős-Pósa property.