Quasimonotone graphs

For any class $\mathcal{C}$ of bipartite graphs, we define quasi-$\mathcal{C}$ to be the class of all graphs $G$ such that every bipartition of $G$ belongs to $\mathcal{C}$. 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.