Permutation bigraphs and interval containments

A bipartite graph with partite sets XX and YY is a permutation bigraph   if there are two linear orderings of its vertices such that xyxy is an edge for x∈Xx∈X and y∈Yy∈Y if and only if xx appears later than yy in the first ordering and earlier than yy in the second ordering. We characterize permutation bigraphs in terms of representations using intervals. We determine which permutation bigraphs are interval bigraphs or indifference bigraphs in terms of the defining linear orderings. Finally, we show that interval containment posets are precisely those whose comparability bigraphs are permutation bigraphs, via a theorem showing that a directed version of interval containment provides no more generality than ordinary interval containment representation of posets.