Algorithms for P 4-comparability Graph Recognition and Acyclic P 4-transitive Orientation

We consider two problems pertaining to P4-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P4-comparability graph and the problem of producing an acyclic P4-transitive orientation of a P4-comparability graph. These problems have been considered by Hoàng and Reed who described O(n4) and O(n5)-time algorithms for their solution respectively, where n is the number of vertices of the input graph. Faster algorithms have recently been presented by Raschle and Simon, and by Nikolopoulos and Palios; the time complexity of these algorithms for either problem is O(n + m2), where m is the number of edges of the graph. In this paper, we describe O(n m)-time and O(n + m)-space algorithms for the recognition and the acyclic P4-transitive orientation problems on P4-comparability graphs. The algorithms rely on properties of the P4-components of a graph, which we establish, and on the efficient construction of the P4-components by means of the BFS-trees of the complement of the graph rooted at each of its vertices, without however explicitly computing the complement. Both algorithms are simple and use simple data structures.