Quasimonotone graphs

TL;DR

This paper introduces quasi-monotone graphs, a generalization of monotone graphs, and provides a polynomial-time recognition algorithm by analyzing their structure, extending bipartite graph concepts to nonbipartite graphs.

Contribution

It defines quasi-monotone graphs based on bipartition classes and develops a polynomial-time recognition algorithm for this new class.

Findings

Quasi-monotone graphs include monotone graphs as a special case.

A structural characterization of quasi-monotone graphs is provided.

A polynomial-time recognition algorithm for quasi-monotone graphs is developed.

Abstract

For any class $\mathcal{C}$ of bipartite graphs, we define quasi-$\cal C$ to be the class of all graphs $G$ such that every bipartition of $G$ belongs to $\cal C$. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.