Two fixed-parameter algorithms for the cocoloring problem

  • Authors:

  • Victor Campos;Sulamita Klein;Rudini Sampaio;Ana Silva

  • Affiliations:

  • Universidade Federal do Ceará, Fortaleza, Brazil;Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil;Universidade Federal do Ceará, Fortaleza, Brazil;Universidade Federal do Ceará, Fortaleza, Brazil

  • Venue:
  • ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
  • Year:
  • 2011

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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 P4-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 P4's. P4-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.