On the Recognition of P4-Comparability Graphs

We consider the problem of recognizing whether a simple undirected graph is a P4-comparability graph. This problem has been considered by Hoàng and Reed who described an O(n)-time algorithm for its solution, where n is the number of vertices of the given graph. Faster algorithms have recently been presented by Raschle and Simon and by Nikolopoulos and Palios; the time complexity of both algorithms is O(n+m), where m is the number of edges of the graph. In this paper, we describe an O(nm)-time, O(n+m)-space algorithm for the recognition of P4-comparability graphs. The algorithm computes the P4s of the input graph G by means of the BFS-trees of the complement of G rooted at each of its vertices, without however explicitly computing the complement of G. Our algorithm is simple, uses simple data structures, and leads to an O(nm)-time algorithm for computing an acyclic P4transitive orientation of a P4-comparability graph.