Algorithms for Unipolar and Generalized Split Graphs

A graph G = (V,E) is a unipolar graph if there exits a partition V = V1 ∪ V2 such that, V1 is a clique and V2 induces the disjoint union of cliques. The complement-closed class of generalized split graphs are those graphs G such that either G or the complement of G is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C5-free (and hence, almost all perfect graphs) are generalized split graphs. In this paper we present a recognition algorithm for unipolar graphs that utilizes a minimal triangulation of the given graph, and produces a partition when one exists. Our algorithm has running time O(nm), where m is the number of edges in a minimal triangulation of the given graph. Generalized split graphs can recognized via this algorithm in O(nm+nm) = O(n) time. We give algorithms on unipolar graphs for finding a maximum independent set and a minimum clique cover in O(n+m) time and for finding a maximum clique and a minimum proper coloring in O(n/ logn), when a unipolar partition is given. These algorithms yield algorithms for the four optimization problems on generalized split graphs that have the same worst-case time bound. We also prove that the perfect code problem is NP-Complete for unipolar graphs.