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.
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© 2021, Society for Industrial and Applied Mathematics.
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Submitted: 12 November 2018
Accepted: 18 January 2021
Published online: 7 April 2021
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Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 648527
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Institute for Basic Science https://doi.org/10.13039/501100010446 : IBS-R029-C1
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National Research Foundation of Korea https://doi.org/10.13039/501100003725 : NRF-2018R1D1A1B07050294
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