Characterizations of undirected 2-quasi best match graphs

Bipartite best match graphs (BMG) and their generalizations arise in mathematical phylogenetics as combinatorial models describing evolutionary relationships among related genes in a pair of species. In this work, we characterize the class of \emph{undirected 2-quasi-BMGs} (un2qBMGs), which form a proper subclass of the $P_6$-free chordal bipartite graphs. We show that un2qBMGs are exactly the class of bipartite graphs free of $P_6$, $C_6$, and the eight-vertex Sunlet$_4$ graph. Equivalently, a bipartite graph $G$ is un2qBMG if and only if every connected induced subgraph contains a ``heart-vertex'' which is adjacent to all the vertices of the opposite color. We further provide a $O(|V(G)|^3)$ algorithm for the recognition of un2qBMGs that, in the affirmative case, constructs a labeled rooted tree that ``explains'' $G$. Finally, since un2qBMGs coincide with the $(P_6,C_6)$-free bi-cographs, they can also be recognized in linear time.