Best Match Graphs with Binary Trees

TL;DR

This paper introduces an efficient algorithm to determine if a best match graph can be explained by a fully resolved binary gene tree, and explores the complexity of editing graphs to achieve this explainability.

Contribution

It presents a near-cubic algorithm for binary-explainability of BMGs, characterizes the structure of such trees, and formulates the graph editing problem as an NP-complete challenge.

Findings

Algorithm efficiently determines binary-explainability of BMGs.

All binary-explainable BMGs are refinements of a unique binary-resolvable tree.

Graph editing to binary-explainability is NP-complete, with an ILP formulation provided.

Abstract

Best match graphs (BMG) are a key intermediate in graph-based orthology detection and contain a large amount of information on the gene tree. We provide a near-cubic algorithm to determine whether a BMG is binary-explainable, i.e., whether it can be explained by a fully resolved gene tree and, if so, to construct such a tree. Moreover, we show that all such binary trees are refinements of the unique binary-resolvable tree (BRT), which in general is a substantial refinement of the also unique least resolved tree of a BMG. Finally, we show that the problem of editing an arbitrary vertex-colored graph to a binary-explainable BMG is NP-complete and provide an integer linear program formulation for this task.