Characterization and recognition of P4-sparse graphs partitionable into k independent sets and ℓ cliques

In this work, we focus on the class of P4-sparse graphs, which generalizes the well-known class of cographs. We consider the problem of verifying whether a P4-sparse graph is a (k,ℓ)-graph, that is, a graph that can be partitioned into k independent sets and ℓ cliques. First, we describe in detail the family of forbidden induced subgraphs for a cograph to be a (k,ℓ)-graph. Next, we show that the same forbidden structures suffice to characterize P4-sparse graphs which are (k,ℓ)-graphs. Finally, we describe how to recognize (k,ℓ)-P4-sparse graphs in linear time by using special auxiliary cographs.