Blocks and Cut Vertices of the Buneman Graph

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

Computing the blocks of a quasi-median graph

Quasi-median graphs are a tool commonly used by evolutionary biologists to visualise the evolution of molecular sequences. As with any graph, a quasi-median graph can contain cut vertices, that is, vertices whose removal disconnect the graph. These vertices induce a decomposition of the graph into blocks, that is, maximal subgraphs which do not contain any cut vertices. Here we show that the special structure of quasi-median graphs can be used to compute their blocks without having to compute the whole graph. In particular we present an algorithm that, for a collection of $n$ aligned sequences of length $m$, can compute the blocks of the associated quasi-median graph together with the information required to correctly connect these blocks together in run time $\mathcal O(n^2m^2)$, independent of the size of the sequence alphabet. Our primary motivation for presenting this algorithm is the fact that the quasi-median graph associated to a sequence alignment must contain all most parsimonious trees for the alignment, and therefore precomputing the blocks of the graph has the potential to help speed up any method for computing such trees.Comment: 17 pages, 2 figure

The polytopal structure of the tight-span of a totally split-decomposable metric

The tight-span of a finite metric space is a polytopal complex that has appeared in several areas of mathematics. In this paper we determine the polytopal structure of the tight-span of a totally split-decomposable (finite) metric. These metrics are a generalization of tree-metrics and have importance within phylogenetics. In previous work, we showed that the cells of the tight-span of such a metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra. Here, we extend these results and show that the tight-span of a totally split-decomposable metric can be broken up into a canonical collection of polytopal complexes whose polytopal structures can be directly determined from the metric. This allows us to also completely determine the polytopal structure of the tight-span of a totally split-decomposable metric. We anticipate that our improved understanding of this structure may lead to improved techniques for phylogenetic inference

Fundamental polytopes of metric trees via parallel connections of matroids

We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010. In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid. We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics, and we characterize the metric trees for which the fundamental polytope is simplicial.Comment: 20 pages, 2 Figures, 1 Table. Exposition improved, references and new results (last section) adde

Uprooted Phylogenetic Networks

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system  Σ(N)Σ(N)  induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems

Injective split systems

A split system $\mathcal S$ on a finite set $X$, $|X|\ge3$, is a set of bipartitions or splits of $X$ which contains all splits of the form $\{x,X-\{x\}\}$, $x \in X$. To any such split system $\mathcal S$ we can associate the Buneman graph $\mathcal B(\mathcal S)$ which is essentially a median graph with leaf-set $X$ that displays the splits in $\mathcal S$. In this paper, we consider properties of injective split systems, that is, split systems $\mathcal S$ with the property that $\mathrm{med}_{\mathcal B(\mathcal S)}(Y) \neq \mathrm{med}_{\mathrm B(\mathcal S)}(Y')$ for any 3-subsets $Y,Y'$ in $X$, where $\mathrm {med}_{\mathcal B(\mathcal S)}(Y)$ denotes the median in $\mathcal B(\mathcal S)$ of the three elements in $Y$ considered as leaves in $\mathcal B(\mathcal S)$. In particular, we show that for any set $X$ there always exists an injective split system on $X$, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph $\mathcal B(\mathcal S)$ needs to become in order for a split system $\mathcal S$ on $X$ to be injective. We do this by introducing a quantity for $|X|$ which we call the injective dimension for $|X|$, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map on $X$ can be represented using Buneman graphs.Comment: 22 pages, 3 figure

The leafage of a chordal graph

The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n - lg n - (1/2) lg lg n + O(1). The proper leafage l*(G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l*(G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.Comment: 19 pages, 3 figure