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Register a new userThe Split Decomposition of a k-Dissimilarity Map
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A k-dissimilarity map on a finite set X is a function D : X choose k rightarrow R assigning a real value to each subset of X with cardinality k, k geq 2. Such functions, also sometimes known as k-way dissimilarities, k-way distances, or k-semimetrics, are of interest in many areas of mathematics, computer science and classification theory, especially 2-dissimilarity maps (or distances) which are a generalisation of metrics. In this paper, we show how regular subdivisions of the kth hypersimplex can be used to obtain a canonical decomposition of a k-dissimilarity map into the sum of simpler k-dissimilarity maps arising from bipartitions or splits of X. In the special case k = 2, this is nothing other than the well-known split decomposition of a distance due to Bandelt and Dress [Adv. Math. 92 (1992), 47-105], a decomposition that is commonly to construct phylogenetic trees and networks. Furthermore, we characterise those sets of splits that may occur in the resulting decompositions of k-dissimilarity maps. As a corollary, we also give a new proof of a theorem of Pachter and Speyer [Appl. Math. Lett. 17 (2004), 615-621] for recovering k-dissimilarity maps from trees.
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Tight-spans of metrics were first introduced by Isbell in 1964 and rediscovered and studied by others, most notably by Dress, who gave them this name. Subsequently, it was found that tight-spans could be defined for more general maps, such as directed metrics and distances, and more recently for diversities. In this paper, we show that all of these tight-spans as well as some related constructions can be defined in terms of point configurations. This provides a useful way in which to study these objects in a unified and systematic way. We also show that by using point configurations we can recover results concerning one-dimensional tight-spans for all of the maps we consider, as well as extend these and other results to more general maps such as symmetric and unsymmetric maps.
Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=knpm 1$. They introduced an intermediate family $mathcal{T}_n^k$ of certain standard Young tableau. Then inverting the sweep map is done by a simple walking algorithm on a $Tin mathcal{T}_n^k$. We find their idea naturally extends for $mathbf{k}^pm$-Dyck paths, and also for $mathbf{k}$-Dyck paths (reducing to $k$-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in $mathcal{T}_mathbf{k}$ of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.
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.
It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller diameter. The argument is due to N. Alon and is a remarkable application of combinatorics and algebra to geometry. This note is purely expository and is accessible for students.
A simple graph is $3$-rigid if its generic bar-joint frameworks in $R^3$ are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal $3$-rigidity of a simple graph which is obtained from the $1$-skeleton of a triangulated torus by the deletion of edges interior to a triangulated disc.
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