Fixed-parameter algorithms for the cocoloring problem

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. It is known that determining the cochromatic number and the split chromatic number, which are respectively the minimum k+ℓ and the minimum max{k,ℓ} such that G is (k,ℓ)-cocolo...

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Published in:Discrete Applied Mathematics Vol. 167; pp. 52 - 60
Main Authors: Campos, Victor, Klein, Sulamita, Sampaio, Rudini, Silva, Ana
Format: Journal Article
Language:English
Published: Elsevier B.V 20.04.2014
Subjects:
[formula omitted]-graphs
Algorithms
Bounded treewidth
Cochromatic number
Cocoloring
Fixed-parameter tractability
Graphs
Mathematical analysis
Partitions
Polynomials
Cochromatic number
Bounded treewidth
Cocoloring
Fixed-parameter tractability
(q,q−4)-graphs
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary: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. It is known that determining the cochromatic number and the split chromatic number, which are respectively the minimum k+ℓ and the minimum max{k,ℓ} such that G is (k,ℓ)-cocolorable, is NP-hard problem. A (q,q−4)-graph is a graph such that every subset of at most q vertices induces at most q−4 distinct P4’s. In 2011, Bravo et al. obtained a polynomial time algorithm to decide if a (5,1)-graph is (k,ℓ)-cocolorable (Bravo et al., 2011). In this paper, we extend this result by obtaining polynomial time algorithms to decide the (k,ℓ)-cocolorability and to determine the cochromatic number and the split chromatic number for (q,q−4)-graphs for every fixed q and for graphs with bounded treewidth. We also obtain a polynomial time algorithm to obtain the maximum (k,ℓ)-cocolorable subgraph of a (q,q−4)-graph for every fixed q. All these algorithms are fixed parameter tractable.
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ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2013.11.010