Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s, k)-polar if there exists a partition A, B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph). Recognizing a polar graph is known to be NP -complete. We provide a polynomial time algorithm for finding a largest polar induced subgraph in cographs (graphs without induced path on four vertices). A characterization of polar cographs in terms of forbidden subgraphs is given. We examine also the monopolar cographs which are the (s, k)-polar cographs where min(s, k) ≤ 1. A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.