Fixed-parameter algorithms for the cocoloring problem
A (k,?) ( k , ? ) -cocoloring of a graph G G is a partition of the vertex set of G G into at most k k independent sets and at most ? ? cliques. It is known that determining the cochromatic number and the split chromatic number, which are respectively the minimum k+? k + ? and the minimum max{k,?} max { k , ? } such that G G is (k,?) ( k , ? ) -cocolorable, is NP-hard problem. A (q,q?4) ( q , q ? 4 ) -graph is a graph such that every subset of at most q q vertices induces at most q?4 q ? 4 distinct P4 P 4 s. In 2011, Bravo et al. obtained a polynomial time algorithm to decide if a (5,1) ( 5 , 1 ) -graph is (k,?) ( k , ? ) -cocolorable (Bravo etal., 2011). In this paper, we extend this result by obtaining polynomial time algorithms to decide the (k,?) ( k , ? ) -cocolorability and to determine the cochromatic number and the split chromatic number for (q,q?4) ( q , q ? 4 ) -graphs for every fixed q q and for graphs with bounded treewidth. We also obtain a polynomial time algorithm to obtain the maximum (k,?) ( k , ? ) -cocolorable subgraph of a (q,q?4) ( q , q ? 4 ) -graph for every fixed q q . All these algorithms are fixed parameter tractable.
cocoloring cochromatic number (q q?4) ( q q?4) ( q q?4) ( q q ? 4 ) bounded treewidth fixed-parameter tractability q ? 4 ) -graphs (q q ? 4 ) (q q ? 4 ) ( q