Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1101.4927

Blocks and Cut Vertices of the Buneman Graph

A. W. W. Dress·K. T. Huber·J. Koolen·V. Moulton
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. W. W. Dress, K. T. Huber, J. Koolen, V. Moulton. 2011-01-25. Blocks and Cut Vertices of the Buneman Graph. https://arxiv.org/abs/1101.4927

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗