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Parity Linkage and the Erdős-Pósa Property of Odd Cycles Through Prescribed Vertices in Highly Connected Graphs

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Book cover Graph-Theoretic Concepts in Computer Science (WG 2015)

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Abstract

We show the following for every sufficiently connected graph G, any vertex subset S of G, and given integer k: there are k disjoint odd cycles in G containing each a vertex of S or there is set X of at most \(3k-3\) vertices such that \(G-X\) does not contain any odd cycle that contains a vertex of S. We prove this via an extension of Kawarabayashi and Reed’s result about parity-k-linked graphs (Combinatorica 29, 215–225). From this result it is easy to deduce several other well known results about the Erdős-Pósa property of odd cycles in highly connected graphs. This strengthens results due to Thomassen (Combinatorica 21, 321–333), and Rautenbach and Reed (Combinatorica 21, 267–278), respectively. Furthermore, we consider algorithmic consequences of our results.

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Authors and Affiliations

  1. Institut für Optimierung und Operations Research, Universität Ulm, Helmholtzstraße 18, 89081, Ulm, Germany

    Felix Joos

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  1. Felix Joos
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Correspondence to Felix Joos .

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  1. TU München Institut für Informatik, Garching, Germany

    Ernst W. Mayr

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Joos, F. (2016). Parity Linkage and the Erdős-Pósa Property of Odd Cycles Through Prescribed Vertices in Highly Connected Graphs. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_24

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  • DOI: https://doi.org/10.1007/978-3-662-53174-7_24

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  • Print ISBN: 978-3-662-53173-0

  • Online ISBN: 978-3-662-53174-7

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