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{\bf Katharina T. Huber, Jacobus Koolen, Vincent Moulton and Andreas Spillner}
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{\bf Characterizing Cell-Decomposable Metrics}
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To a finite metric space $(X,d)$ one can associate the so called {\em
tight-span} $T(d)$ of $d$, that is, a canonical metric space
$(T(d),d_\infty)$ into which $(X,d)$ isometrically embeds and which may be
thought of as the abstract convex hull of $(X,d)$. Amongst other
applications, the tight-span of a finite metric space has been used to
decompose and classify finite metrics, to solve instances of the
server and multicommodity flow problems, and to perform evolutionary
analyses of molecular data. To better understand the structure of
$(T(d),d_\infty)$ the concept of a {\em cell-decomposable} metric was
recently introduced, a metric whose associated tight-span can be
decomposed into simpler tight-spans. Here we show that
cell-decomposable metrics and {\em totally split-decomposable metrics}
--- a class of metrics commonly applied within phylogenetic analysis
--- are one and the same thing, and also provide some additional
characterizations of such metrics.
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