Fixed-parameter algorithms for the cocoloring problem - Campos, Victor; Klein, Sulamita; Sampaio, Rudini; Silva, Ana (Discrete Applied Mathematics, 2014)
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.