Blocks and Cut Vertices of the Buneman Graph
A.W.W. Dress, K.T. Huber, J. Koolen, V. Moulton

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 of bipartitions of some finite set of cardinality at least 2, one can associate to a canonical -labeled graph , 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 , 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 is a tree if and only if any two splits in are compatible.
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Taxonomy
TopicsAlgorithms and Data Compression · Genomic variations and chromosomal abnormalities · Genome Rearrangement Algorithms
