Forbidden configurations and dominating bicliques in undirected 2-quasi best match graphs

TL;DR

This paper studies the structural properties of undirected graphs derived from 2-quasi best match graphs in phylogenetics, revealing forbidden configurations and the presence of dominating bicliques.

Contribution

It characterizes the forbidden induced subgraphs in un-2qBMGs and demonstrates the existence of vertex decompositions with dominating bicliques.

Findings

Un-2qBMGs contain no induced paths or cycles of length ≥6.

Existence of specific vertex decompositions with dominating bicliques.

Structural insights into phylogenetic relatedness graphs.

Abstract

2-quasi best match graphs (2-qBMGs) are directed graphs that capture a notion of close relatedness in phylogenetics. Here, we investigate the undirected underlying graph of a 2-qBMG (un-2qBMG) and show that they contain neither a path $P_l$ nor a cycle $C_l$ of length $l\geq 6$ as an induced subgraph. This property guarantees the existence of specific vertex decompositions with dominating bicliques that provide further insights into their structure.