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Partitioning sparse graphs into an independent set and a graph with bounded size components

Abstract : We study the problem of partitioning the vertex set of a given graph so that each part induces a graph with components of bounded order; we are also interested in restricting these components to be paths. In particular, we say a graph G admits an (I, O k)-partition if its vertex set can be partitioned into an independent set and a set that induces a graph with components of order at most k. We prove that every graph G with mad(G) < 5 2 admits an (I, O 3)-partition. This implies that every planar graph with girth at least 10 can be partitioned into an independent set and a set that induces a graph whose components are paths of order at most 3. We also prove that every graph G with mad(G) < 8k 3k+1 = 8 3 1 − 1 3k+1 admits an (I, O k)-partition. This implies that every planar graph with girth at least 9 can be partitioned into an independent set and a set that induces a graph whose components are paths of order at most 9.
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https://hal.archives-ouvertes.fr/hal-02990588
Contributor : Pascal Ochem Connect in order to contact the contributor
Submitted on : Monday, November 16, 2020 - 10:24:09 AM
Last modification on : Monday, October 11, 2021 - 1:24:09 PM
Long-term archiving on: : Wednesday, February 17, 2021 - 6:31:47 PM

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Ilkyoo Choi, François Dross, Pascal Ochem. Partitioning sparse graphs into an independent set and a graph with bounded size components. Discrete Mathematics, Elsevier, 2020, 343 (8), pp.111921. ⟨10.1016/j.disc.2020.111921⟩. ⟨hal-02990588⟩

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