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Finding Induced Paths of Given Parity in Claw-Free Graphs

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Book cover Graph-Theoretic Concepts in Computer Science (WG 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5911))

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Abstract

The Parity Path problem is to decide if a given graph G contains both an odd length and an even length induced path between two specified vertices s and t. In the related problems Odd Induced Path and Even Induced Path, the goal is to determine whether an induced path of odd, respectively even, length between two specified vertices exists. Although all three problems are NP-complete in general, we show that they can be solved in \({\cal O}(n^5)\) time for the class of claw-free graphs. Two vertices s and t form an even pair in G if every induced path from s to t in G has even length. Our results imply that the problem of deciding if two specified vertices of a claw-free graph form an even pair, as well as the problem of deciding if a given claw-free graph has an even pair, can be solved in \({\cal O}(n^5)\) time and \({\cal O}(n^7)\) time, respectively. We also show that we can decide in \({\cal O}(n^7)\) time whether a claw-free graph has an induced cycle of given parity through a specified vertex.

This work has been supported by EPSRC (EP/D053633/1) and the Actions de Recherche Concertées (ARC) fund of the Communauté française de Belgique.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, England

    Pim van ’t Hof & Daniël Paulusma

  2. Computer Science Department, Université Libre de Bruxelles, Boulevard du Triomphe CP212, B-1050, Brussels, Belgium

    Marcin Kamiński

Authors
  1. Pim van ’t Hof
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  2. Marcin Kamiński
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  3. Daniël Paulusma
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Editor information

Editors and Affiliations

  1. CNRS - LIRMM, 161 rue Ada, 34392, Montpellier Cedex 5, France

    Christophe Paul

  2. LIAFA, case 7014, Université Paris Diderot, 75205, Paris Cedex 13, France

    Michel Habib

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van ’t Hof, P., Kamiński, M., Paulusma, D. (2010). Finding Induced Paths of Given Parity in Claw-Free Graphs. In: Paul, C., Habib, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2009. Lecture Notes in Computer Science, vol 5911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11409-0_30

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  • DOI: https://doi.org/10.1007/978-3-642-11409-0_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11408-3

  • Online ISBN: 978-3-642-11409-0

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