One of the main problems in phylogenetics is to find good approximations of metrics by weighted trees. As an aid to solving this problem, it could be tempting to consider optimal realizations of metrics -the guiding principle being that, the (necessarily unique) optimal realization of a tree metric is the weighted tree that realizes this metric. And, although optimal realizations of arbitrary metrics are, in general, not trees, but rather weighted networks, one could still hope to obtain a phylogenetically informative representation of a given metric, maybe even more informative than the best approximating tree. However, optimal realizations are not only difficult to compute, they may also be non-unique. Here we focus on one possible way out of this dilemma: hereditarily optimal realizations. These are essentially unique, and can be described in a rather explicit way. In this paper, we recall what a hereditarily optimal realization of a metric is and how it is related to the 1-skeleton of the tight span of that metric, and we investigate under what conditions it coincides with this 1-skeleton. As a consequence, we will show that hereditarily optimal realizations for consistent metrics, a large class of phylogentically relevant metrics, can be computed in a straight-forward fashion.
Given a metric d on a finite set X , a realization of d is a weighted graph G = (V, E, w: E → R >0 ) with X ⊆ V such that for all x, y ∈ X the length of any shortest path in G between x and y equals d (x, y). In this paper we consider two special kinds of realizations, optimal realizations and hereditarily optimal realizations, and their relationship with the so-called tight span. In particular, we present an infinite family of metrics {d k } k≥1 , and-using a new characterization for when the so-called underlying graph of a metric is an optimal realization that we also present-we prove that d k has (as a function of k) exponentially many optimal realizations with distinct degree sequences. We then show that this family of metrics provides counter-examples to a conjecture made by Dress in 1984 concerning the relationship between optimal realizations and the tight span, and a negative reply to a question posed by Althöfer in 1988 on the relationship between optimal and hereditarily optimal realizations.
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