A characterization of dissimilarity families of trees

Baldisserri, Agnese; Rubei, Elena

doi:10.1016/j.dam.2016.12.007

Cited by 2 publications

(3 citation statements)

References 17 publications

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“…Consider the 13 × 35 matrix whose columns are indexed by the size 4 subsets of [7] (ordered lexicographically: {1, 2, 3, 4} < {1, 2, 3, 5} < • • • < {4, 5, 6, 7}), whose first 10 rows are the images under d4 of the split metrics defined by the edges in G, and whose last three rows are the images of the split metrics defined by the edges Ei :…”

Section: Resolution Of Pachter and Speyer's Second Questionmentioning

confidence: 99%

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Point configurations, phylogenetic trees, and dissimilarity vectors

Caminata¹,

Giansiracusa²,

Moon³

et al. 2021

Proc. Natl. Acad. Sci. U.S.A.

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In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

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Section: Resolution Of Pachter and Speyer's Second Questionmentioning

confidence: 99%

“…There have been numerous papers studying other aspects of Pachter and Speyer's higher dissimilarity maps as well (e.g., refs. [6][7][8][9][10][11][12][13].…”

mentioning

confidence: 99%

Point configurations, phylogenetic trees, and dissimilarity vectors

Caminata¹,

Giansiracusa²,

Moon³

et al. 2021

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

show abstract

“…In order to study the families of multiweights of nonnegative-weighted trees with set of leaves equal to [n], we need to recall some notation and some facts from [1].…”

Section: Trees With Labels Only On the Leavesmentioning

confidence: 99%

Graphlike families of multiweights

Baldisserri¹,

Rubei²

2016

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Let G = (G, w) be a weighted graph, that is, a graph G endowed with a function w from the edge set of G to the set of real numbers; for any subset S of the vertex set of G, we define D S (G) to be the minimum of the weights of the subgraphs of G whose vertex set contains S; we call D S (G) a multiweight of G.Let X be a finite set and let {D S } S⊂X, ♯S≥2 be a family of positive real numbers. We find necessary and sufficient conditions for the family to be the family of multiweights of a positiveweighted graph with vertex set X. Moreover we study the analogous problem for trees. Finally, we find a criterion to say if there exists a nonnegative-weighted tree T with leaf set X and such that D S (T ) = D S for any S ⊂ X.

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A characterization of dissimilarity families of trees

Abstract: Let T = (T, w) be a weighted finite tree with leaves 1, .

Cited by 2 publications

References 17 publications

Point configurations, phylogenetic trees, and dissimilarity vectors

Point configurations, phylogenetic trees, and dissimilarity vectors

Graphlike families of multiweights

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