Arc-Completion of 2-Colored Best Match Graphs to Binary-Explainable Best Match Graphs

TL;DR

This paper studies the structure of best match graphs in phylogenetics, proving properties of least resolved trees and providing a polynomial-time algorithm for completing two-colored BMGs to binary-explainable forms.

Contribution

It introduces a characterization of least resolved trees for two-colored BMGs and presents a polynomial-time method for arc completion to binary-explainable BMGs.

Findings

Inner edge contraction preserves least resolved tree property.

Characterization of least resolved trees for two-colored BMGs.

Polynomial-time algorithm for arc completion to binary-explainable BMGs.

Abstract

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.