Minimal obstructions to [formula omitted]-polarity in cographs

Abstract

Let s,k be non negative integers. A graph G is (s,k)-polar if its vertex set admits a partition (A,B) such that A induces a complete multipartite graph with at most s parts, and B induces a disjoint union of at most k cliques with no other edges. A notable case is that of (1,1)-polar graphs, which are exactly split graphs. The concepts of (s,∞)-, (∞,k)- and (∞,∞)-polar graphs can be analogously defined. A graph is a cograph if it does not contain P4 as an induced subgraph.Ekim, Mahadev and de Werra first investigated polar cographs, obtaining forbidden induced subgraph characterizations for (∞,∞)-polar cographs, and also for the union of (1,∞)- and (∞,1)-polar cographs. So far, for fixed s and k, complete lists of minimal(s,k)-polar cograph obstructions are known only for the pairs (1,1),(1,2),(2,1) and (2,2).In this paper we find the list of cograph minimal (s,1)-polar obstructions for any fixed positive integer s, as well as the list of cograph minimal (∞,1)-polar obstructions. By considering the complement of such graphs, these results extend to (1,k)- and (1,∞)-polar cographs.