Blocks and Cut Vertices of the Buneman Graph

TL;DR

This paper investigates the structural properties of the Buneman graph, focusing on its cut vertices and blocks, and generalizes the condition under which it forms a tree based on split compatibility.

Contribution

It provides new insights into the cut vertices and blocks of the Buneman graph, extending the understanding of when the graph is a tree based on split compatibility.

Findings

Characterization of cut vertices in the Buneman graph

Analysis of blocks and 2-connected components

Generalization of tree conditions based on split compatibility

Abstract

Given a set $\Sg$ of bipartitions of some finite set $X$ of cardinality at least 2, one can associate to $\Sg$ a canonical $X$-labeled graph $\B(\Sg)$, called the Buneman graph. This graph has several interesting mathematical properties - for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as a tool in studies of DNA sequences gathered from populations. In this paper, we present some results concerning the {\em cut vertices} of $\B(\Sg)$, i.e., vertices whose removal disconnect the graph, as well as its {\em blocks} or 2-{\em connected components} - results that yield, in particular, an intriguing generalization of the well-known fact that $\B(\Sg)$ is a tree if and only if any two splits in $\Sg$ are compatible.