Parallel algorithms for P4-comparability graphs

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 such a graph. Sequential algorithms for these problems have been presented by Hoàng and Reed and very recently by Raschle and Simon, and by Nikolopoulos and Palios. In this paper, we establish properties of P4-comparability graphs which allow us to describe parallel algorithms for the recognition and orientation problems on this class of graphs; for a graph on n vertices and m edges, our algorithms run in O(log n) time and require O(nm/ log n) processors on the CREW PRAM model. Since the currently fastest sequential algorithms for these problems run in O(nm) time, our algorithms are cost-efficient; moreover, to the best of our knowledge, this is the first attempt to introduce parallelization in problems involving P4-comparability graphs. Our approach relies on the parallel computation and proper orientation of the P4-components of the input graph.