Polarity on H-split graphs

Abstract

Given nonnegative integers, s and k, an (s, k)-polar partition of a graph G is a partition (A, B) of VG such that G[A] and ̅G[B] are complete multipartite graphs with at most s and k parts, respectively. If s or k is replaced by ∞, it means that there is no restriction on the number of parts of G[A] or ̅G[B], respectively. A split graph is a graph admitting a (1, 1)-polar partition. A graph is said to be unipolar or monopolar if its vertex set admits an (∞, ∞)-polar partition (A, B) such that A is a clique or an independent set, respectively.Naturally, most problems related to polar partitions are trivial on split graphs, even when some of them are very hard in general. In this work, we present some results related to polar partitions on two graph classes generalizing split graphs. Our main results include efficient algorithms to decide whether a graph on these classes admits such partitions, as well as upper bounds for the order of minimal (s, k)-polar obstructions on such graph families for any s and k (even if s or k is ∞).