# Comparability and Cocomparability Bigraphs

## TL;DR

This paper introduces bipartite analogues of comparability and cocomparability graphs, showing they are equivalent and providing characterizations and recognition algorithms for these cocomparability bigraphs.

## Contribution

It defines cocomparability bigraphs, characterizes them via forbidden structures and orderings, and develops a polynomial-time recognition algorithm.

## Key findings

- Cocomparability bigraphs are characterized by the absence of edge-asteroids.
- They are exactly the bipartite graphs with no edge-asteroids.
- A polynomial-time recognition algorithm for cocomparability bigraphs is provided.

## Abstract

We propose bipartite analogues of comparability and cocomparability graphs. Surprizingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderings, forbidden substructures, and orientations of their complements. In particular, we prove that cocomparability bigraphs are precisely those bipartite graphs that do not have edge-asteroids; this is analogous to Gallai's structural characterization of cocomparability graphs by the absence of (vertex-) asteroids. Our characterizations imply a robust polynomial-time recognition algorithm for the class of cocomparability bigraphs. Finally, we also discuss a natural relation of cocomparability bigraphs to interval containment bigraphs, resembling a well-known relation of cocomparability graphs to interval graphs.