# Recognizing Treelike *k*-Dissimilarities

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## Abstract

A *k*-dissimilarity *D* on a finite set *X*, |*X*| *≥ k*, is a map from the set of size *k* subsets of *X* to the real numbers. Such maps naturally arise from edgeweighted trees *T* with leaf-set *X*: Given a subset *Y* of *X* of size *k*, *D*(*Y* ) is defined to be the total length of the smallest subtree of *T* with leaf-set *Y* . In case *k* = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called “4-point condition”. However, in case *k >* 2 Pachter and Speyer (2004) recently posed the following question: Given an arbitrary *k*-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for *k ≥* 3 a *k*-dissimilarity on a set *X* arises from a tree if and only if its restriction to every 2 *k*-element subset of *X* arises from some tree, and that 2 *k* is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a *k*-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.

### Keywords

*k*-dissimilarity Phylogenetic tree Dissimilarity Metric 4-point condition Ultrametric condition Equidistant tree

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