Recognizing bipolarizable and P4-simplicial graphs

Abstract

Hoàng and Reed defined the classes of Raspail (also known as Bipolarizable) and P4-simplicial graphs, both of which are perfectly orderable, and proved that they admit polynomial-time recognition algorithms [16]. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(nm) time, where n and m are the numbers of vertices and of edges of the input graph. In particular, we prove properties and show that we can produce bipolarizable and P4-simplicial orderings on the vertices of a graph G, if such orderings exist, working only on P3s that participate in P4s of G. The proposed recognition algorithms are simple, use simple data structures and require O(n + m) space. Moreover, we present a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs and some preliminary results on forbidden subgraphs for the class of P4-simplicial graphs. © Springer-Verlag Berlin Heidelberg 2003.