Comparability digraphs: An analogue of comparability graphs

TL;DR

This paper introduces comparability digraphs as an extension of comparability graphs, providing structural characterizations, a generalized Triangle Lemma, and an efficient recognition algorithm for semicomplete cases.

Contribution

It defines comparability digraphs, extends key concepts like implication classes, and develops a recognition algorithm with structural characterizations.

Findings

Characterization of comparability digraphs via knotting graphs

Generalization of the Triangle Lemma to semicomplete digraphs

An $ ext{O}(n^3)$ recognition algorithm for semicomplete comparability digraphs

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 are a prototype of comparability digraphs. One instrumental technique for analyzing the structure of comparability graphs is the Triangle Lemma for graphs. We generalize the Triangle Lemma to semicomplete digraphs. Using the Triangle Lemma for semicomplete digraphs we prove that if an implication class of a semicomplete digraph contains no circuit of length 2 then it contains no circuit at all. We also use it to device an $\mathcal{O}(n^3)$ time recognition algorithm for semicomplete comparability digraphs where $n$ is the number of vertices of the input digraph. The correctness of the algorithm implies a characterization for semicomplete comparability digraphs, akin to that for comparability graphs.