# Blocks and Cut Vertices of the Buneman Graph

Dress, AWM, Huber, KT, Koolen, J and Moulton, V (2011) Blocks and Cut Vertices of the Buneman Graph. SIAM Journal on Discrete Mathematics, 25 (4). pp. 1902-1919.

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## Abstract

Given a set $\Sigma$ of bipartitions of some finite set $X$ of cardinality at least $2$, one can associate to $\Sigma$ a canonical $X$-labeled graph $\mathcal{B}(\Sigma)$, 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 cut vertices of $\mathcal{B}(\Sigma)$, i.e., vertices whose removal disconnect the graph, as well as its blocks or $2$-connected components—results that yield, in particular, an intriguing generalization of the well-known fact that $\mathcal{B}(\Sigma)$ is a tree if and only if any two splits in $\Sigma$ are compatible

Item Type: Article Faculty of Science > School of Computing Sciences http://dx.doi.org/10.1137/090764360 Rhiannon Harvey 09 Jan 2012 10:25 21 Apr 2020 16:31 https://ueaeprints.uea.ac.uk/id/eprint/35935 10.1137/090764360

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