Antipodal Metrics and Split Systems

@article{Dress2002AntipodalMA,
  title={Antipodal Metrics and Split Systems},
  author={Andreas W. M. Dress and Katharina T. Huber and Vincent Moulton},
  journal={Eur. J. Comb.},
  year={2002},
  volume={23},
  pages={187-200}
}
Recall that a metric d on a finite set X is called antipodal if there exists a map ?: X?X: x?? __x so that d(x, __ x) =d(x,y ) +d(y, __ x) holds for all x,y?X. Antipodal metrics canonically arise as metrics induced on specific weighted graphs, although their abundance becomes clearer in light of the fact that any finite metric space can be isometrically embedded in a more or less canonical way into an antipodal metric space called its full antipodal extension.In this paper, we examine in some… 
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References

SHOWING 1-10 OF 25 REFERENCES
A canonical decomposition theory for metrics on a finite set
Totally Split-Decomposable Metrics of Combinatorial Dimension Two
Abstract. The combinatorial dimension of a metric space (X; d), denoted by dimcombin(d), arises naturally in the subject of T-theory, and, in case X is finite, corresponds with the (topological)
On the tight span of an antipodal graph
An Exceptional Split Geometry
Abstract. In view of results obtained in split decomposition theory, it is of some interest to investigate the structure of weakly compatible split systems. A particular class of such split systems —
Antipodal graphs and oriented matroids
On circuits and pancyclic line graphs
TLDR
It is shown that the bound n - 1 - p(n) can be decreased to (2n + 1)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.
Handbook of Combinatorics
Part 1 Structures: graphs - basic graph theory - paths and circuits, J.A. Bondy, connectivity and network flows, A. Frank, matchings and extensions, W.R. Pulleyblank, colouring, stable sets and
Geometry of cuts and metrics
TLDR
This book draws from the interdisciplinarity of these fields as it gathers methods and results from polytope theory, geometry of numbers, probability theory, design and graph theory around two objects, cuts and metrics.
Six Points Suffice: How to Check for Metric Consistency
TLDR
This paper gives a six-point characterization of consistent metrics amongst the totally decomposable metrics, and indicates that these metrics are of fundamental importance in the analysis of distance tables.
Hereditarily Optimal Realizations: Why are they Relevant in Phylogenetic Analysis, and how does one Compute them
TLDR
Hereditarily optimal realizations are defined, some of their properties are discussed, and it is indicated in particular why, due to recent results on the so-called T-construction of a metric space, it is a straight forward task to compute these realizations for a large class of phylogentically relevant metrics.
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