Forbidden paths and cycles in the undirected underlying graph of a 2-quasi best match graph

TL;DR

This paper proves that the undirected graph underlying a 2-quasi best match graph cannot contain certain induced paths or cycles, enabling new analysis techniques based on known properties of such graphs.

Contribution

It establishes the absence of specific induced subgraphs in 2-qBMGs and introduces new results on dominating bicliques and vertex decompositions.

Findings

Undirected underlying graphs of 2-qBMGs are $P_6$ and $C_6$ free.

New polynomial algorithms for analyzing 2-qBMGs.

Characterization of vertex decompositions in 2-qBMGs.

Abstract

The undirected underlying graph of a 2-quasi best match graph (2-qBMG) is proven not to contain any induced graph isomorphic to $P_6$ or $C_6$. This new feature allows for the investigation of 2-BMGs further by exploiting the numerous known results on $P_6$ and $C_6$ free graphs together with the available polynomial algorithms developed for their studies. In this direction, there are also some new contributions about dominating bicliques and certain vertex decompositions of the undirected underlying graph of a 2-qBMG.