Characterizations of undirected 2-quasi best match graphs

TL;DR

This paper characterizes undirected 2-quasi best match graphs as a subclass of bipartite graphs defined by forbidden subgraphs, providing recognition algorithms and structural insights relevant to phylogenetics.

Contribution

It provides a complete characterization of un2qBMGs via forbidden subgraphs and develops recognition algorithms, linking them to bi-cographs and phylogenetic models.

Findings

Un2qBMGs are exactly the bipartite graphs free of P6, C6, and Sunlet4.

A cubic time recognition algorithm for un2qBMGs is proposed.

Un2qBMGs coincide with (P6,C6)-free bi-cographs, allowing linear time recognition.

Abstract

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.