Comparability digraphs: An analogue of comparability graphs

Abstract

Comparability graphs are a popular class of graphs. We introduce as the digraph analogue of comparability graphs the class of comparability digraphs. We show that many concepts such as implication classes and the knotting graph for a comparability graph can be naturally extended to a comparability digraph. We give a characterization of comparability digraphs in terms of their knotting graphs.Semicomplete comparability digraphs contain in a sense all comparability graphs. One instrumental technique for analyzing the structure of comparability graphs is the Triangle Lemma for graphs (cf. Golumbic (1980) [7]). Using Triangle Lemma, Golumbic proved that a graph is a comparability graph if and only if no implication class of the graph is equal to its inverse. We give examples to show that a similar statement does not hold for general digraphs. We prove that it holds for semicomplete digraphs, that is, a semicomplete digraph is a comparability digraph if and only if no implication class of the digraph is equal to its inverse. The proof is done by generalizing the Triangle Lemma for graphs to semicomplete digraphs. This yields an O(n3)-time recognition algorithm for semicomplete comparability digraphs where n is the number of vertices of the input digraph. The correctness of the algorithm also implies several other characterizations for semicomplete comparability digraphs, akin to those for comparability graphs.