Quasi-Best Match Graphs

TL;DR

Quasi-best match graphs (qBMGs) are a class of directed, vertex-colored graphs modeling restricted evolutionary relationships, with polynomial recognition algorithms and characterizations linking them to phylogenetic trees.

Contribution

This paper introduces qBMGs, characterizes them, and provides polynomial-time recognition algorithms, extending the understanding of best match graphs in phylogenetics.

Findings

qBMGs are hereditary directed, vertex-colored graphs.

Recognition algorithms for qBMGs are polynomial-time.

Two-colored qBMGs satisfy local conditions like bi-transitivity and hierarchy-like neighborhoods.

Abstract

Quasi-best match graphs (qBMGs) are a hereditary class of directed, properly vertex-colored graphs. They arise naturally in mathematical phylogenetics as a generalization of best match graphs, which formalize the notion of evolutionary closest relatedness of genes (vertices) in multiple species (vertex colors). They are explained by rooted trees whose leaves correspond to vertices. In contrast to BMGs, qBMGs represent only best matches at a restricted phylogenetic distance. We provide characterizations of qBMGs that give rise to polynomial-time recognition algorithms and identify the BMGs as the qBMGs that are color-sink-free. Furthermore, two-colored qBMGs are characterized as directed graphs satisfying three simple local conditions, two of which have appeared previously, namely bi-transitivity in the sense of Das et al. (2021) and a hierarchy-like structure of out-neighborhoods, i.e., $N(x)\cap N(y)\in\{N(x),N(y),\emptyset\}$ for any two vertices $x$ and $y$. Further results characterize qBMGs that can be explained by binary phylogenetic trees.