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
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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. 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.
AbstractList A (k,[ell])(k,[ell])-cocoloring of a graph GG is a partition of the vertex set of GG into at most kk independent sets and at most [ell][ell] cliques. It is known that determining the cochromatic number and the split chromatic number, which are respectively the minimum k+[ell]k+[ell] and the minimum max{k,[ell]}max{k,[ell]} such that GG is (k,[ell])(k,[ell])-cocolorable, is NP-hard problem. A (q,q-4)(q,q-4)-graph is a graph such that every subset of at most qq vertices induces at most q-4q-4 distinct P4P4's. In 2011, Bravo et al. obtained a polynomial time algorithm to decide if a (5,1)(5,1)-graph is (k,[ell])(k,[ell])-cocolorable (Bravo et al., 2011). In this paper, we extend this result by obtaining polynomial time algorithms to decide the (k,[ell])(k,[ell])-cocolorability and to determine the cochromatic number and the split chromatic number for (q,q-4)(q,q-4)-graphs for every fixed qq and for graphs with bounded treewidth. We also obtain a polynomial time algorithm to obtain the maximum (k,[ell])(k,[ell])-cocolorable subgraph of a (q,q-4)(q,q-4)-graph for every fixed qq. All these algorithms are fixed parameter tractable.
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.
Author Campos, Victor
Klein, Sulamita
Silva, Ana
Sampaio, Rudini
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  surname: Silva
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  organization: Universidade Federal do Ceará, Fortaleza, Brazil
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Keywords Cochromatic number
Bounded treewidth
Cocoloring
Fixed-parameter tractability
(q,q−4)-graphs
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Snippet 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...
A (k,[ell])(k,[ell])-cocoloring of a graph GG is a partition of the vertex set of GG into at most kk independent sets and at most [ell][ell] cliques. It is...
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SubjectTerms [formula omitted]-graphs
Algorithms
Bounded treewidth
Cochromatic number
Cocoloring
Fixed-parameter tractability
Graphs
Mathematical analysis
Partitions
Polynomials
Title Fixed-parameter algorithms for the cocoloring problem
URI https://dx.doi.org/10.1016/j.dam.2013.11.010
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