  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.
