Algorithms for P4-Comparability Graph Recognition and Acyclic P4-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.