Non-edge orientation and vertex ordering characterizations of some classes of bigraphs

Abstract

Chordal graphs, permutation graphs, and interval graphs are among many classes of graphs which can be characterized by the existence of certain acyclic orientations and vertex orderings. These types of characterizations exist for some of their bipartite analogues such as chordal bipartite graphs and bipartite permutation graphs. Chvátal proved that a bipartite graph G is chordal bipartite if and only if the complement G¯ of G has a vertex ordering ≺ such that for every induced path abcd in G¯, a≺b implies c≺d. Recently, Le proved that a bipartite graph G is a permutation graph if and only if G¯ admits an acyclic orientation such that for every induced path abcd in G¯, ab is an oriented edge if and only if cd is. Interestingly these orientation and vertex ordering characterizations are stated on the complements of bipartite graphs. We show that interval bigraphs and interval containment bigraphs also admit similar characterizations in terms of vertex orderings and acyclic orientations of their complements.