On the Grundy number of graphs with few P4's

  • Authors:

  • J. Araujo;C. Linhares Sales

  • Affiliations:

  • ParGO Research Group, Federal University of Ceará, Campus do Pici, Bloco 910, 60455-760 - Fortaleza, Brazil and Projet Mascotte I3S (CNRS & Univ. de Nice) and INRIA, INRIA Sophia Antipolis, 2 ...;ParGO Research Group, Federal University of Ceará, Campus do Pici, Bloco 910, 60455-760 - Fortaleza, Brazil

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

The Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P"4-free graph and NP-hard if G is a P"5-free graph. In this article, we define a new class of graphs, the fat-extended P"4-laden graphs, and we show a polynomial time algorithm to determine the Grundy number of any graph in this class. Our class intersects the class of P"5-free graphs and strictly contains the class of P"4-free graphs. More precisely, our result implies that the Grundy number can be computed in polynomial time for any graph of the following classes: P"4-reducible, extended P"4-reducible, P"4-sparse, extended P"4-sparse, P"4-extendible, P"4-lite, P"4-tidy, P"4-laden and extended P"4-laden, which are all strictly contained in the fat-extended P"4-laden class.