Partitioning extended P4-laden graphs into cliques and stable sets

A (k,@?)-cocoloring of a graph is a partition of its vertex set into at most k stable sets and at most @? cliques. It is known that deciding if a graph is (k,@?)-cocolorable is NP-complete. A graph is extended P"4-laden if every induced subgraph with at most six vertices that contains more than two induced P"4@?s is {2K"2,C"4}-free. Extended P"4-laden graphs generalize cographs, P"4-sparse and P"4-tidy graphs. In this paper, we obtain a linear time algorithm to decide if, given k,@?>=0, an extended P"4-laden graph is (k,@?)-cocolorable. Consequently, we obtain a polynomial time algorithm to determine the cochromatic number and the split chromatic number of an extended P"4-laden graph. Finally, we present a polynomial time algorithm to find a maximum induced (k,@?)-cocolorable subgraph of an extended P"4-laden graph, generalizing the main results of Bravo et al. (2011) [4] and Demange et al. (2005) [5].