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Two Fixed-Parameter Algorithms for the Cocoloring Problem

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Algorithms and Computation (ISAAC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7074))

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Abstract

A (k,ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent sets and at most ℓ cliques. Given a graph G and integers k and ℓ, the Cocoloring Problem is the problem of deciding if G has a (k,ℓ)-cocoloring. It is known that determining the cochromatic number (the minimum k + ℓ such that G is (k,ℓ)-cocolorable) is NP-hard [24]. In 2011, Bravo et al. obtained a polynomial time algorithm for P 4-sparse graphs [8]. In this paper, we generalize this result by obtaining a polynomial time algorithm for (q,q − 4)-graphs for every fixed q, which are the graphs such that every subset of at most q vertices induces at most q − 4 induced P 4’s. P 4-sparse graphs are (5,1)-graphs. Moreover, we prove that the cocoloring problem is FPT when parameterized by the treewidth tw(G) or by the parameter q(G), defined as the minimum integer q ≥ 4 such that G is a (q,q − 4)-graph.

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Authors and Affiliations

  1. Universidade Federal do Ceará, Fortaleza, Brazil

    Victor Campos, Rudini Sampaio & Ana Silva

  2. Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Sulamita Klein

Authors
  1. Victor Campos
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  2. Sulamita Klein
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  3. Rudini Sampaio
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  4. Ana Silva
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Editor information

Editors and Affiliations

  1. Chuo University, Kasuga, Bunkyo-ku, 112-8551, Tokyo, Japan

    Takao Asano

  2. Gunma University, 1-5-1 Tenjin-Cho, 376-8515, Kiryu-Shi, Japan

    Shin-ichi Nakano

  3. Japan Advanced Institute of Science and Technology, 1-1 Asahidai, 923-1292, Nomi, Ishikawa, Japan

    Yoshio Okamoto

  4. Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, 152-8552, Tokyo, Japan

    Osamu Watanabe

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© 2011 Springer-Verlag Berlin Heidelberg

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Campos, V., Klein, S., Sampaio, R., Silva, A. (2011). Two Fixed-Parameter Algorithms for the Cocoloring Problem. In: Asano, T., Nakano, Si., Okamoto, Y., Watanabe, O. (eds) Algorithms and Computation. ISAAC 2011. Lecture Notes in Computer Science, vol 7074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25591-5_65

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  • DOI: https://doi.org/10.1007/978-3-642-25591-5_65

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25590-8

  • Online ISBN: 978-3-642-25591-5

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