Recognition and Orientation Algorithms for P4-Comparability Graphs

Abstract

We consider two problems pertaining to P ₄-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P ₄-comparability graph and the problem of producing an acyclic P ₄-transitive orientation of a P ₄-comparability graph. These problems have been considered by Hoáng and Reed who described O(n ⁴) and O(n ⁵)-time algorithms for their solution respectively, where n is the number of vertices of the given graph. Recently, Raschle and Simon described O(n + m ²)-time algorithms for these problems, where m is the number of edges of the graph. In this paper, we describe difierent O(n + m ²)-time algorithms for the recognition and the acyclic P ₄-transitive orientation problems on P ₄- comparability graphs. Instrumental in these algorithms are structural relationships of the P ₄-components of a graph, which we establish and which are interesting in their own right. Our algorithms are simple, use simple data structures, and have the advantage over those of Raschle and Simon in that they are non-recursive, require linear space and admit effcient parallelization.