Tree Reconstruction from Triplet Cover Distances

Huber, Katharina T.; Steel, Mike

doi:10.37236/3388

Cited by 5 publications

(11 citation statements)

References 10 publications

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“…These, and other results obtained along the way complement recent work into phylogenetic ‘lasso’ sets (Dress et al. 2012 ; Huber and Steel 2014 ), as well as a Hall-type characterization of the median function on trees in Dress and Steel ( 2009 ).…”

Section: Introductionsupporting

confidence: 81%

“…( 2004 ), Proposition 3.4] (a graph is called a 2d-tree if there exists an ordering of V such that and, for the vertex has degree 2 in the subgraph of G induced by ). So Theorem 1 can be used to strengthen Theorem 1 of Huber and Steel ( 2014 ).…”

Section: A Characterization Of Minimum Triplet Coversmentioning

confidence: 89%

“…Second, the corollary gives an independent proof of Dress et al. ( 2012 ), Theorem 7 and Huber and Steel ( 2014 ), Theorem 1 which state that pointed triplet covers and stable triplet covers are shellable, respectively.…”

Section: Introductionmentioning

confidence: 90%

“…Moreover, there are various ways to construct triplet covers that are minimum [for example, ‘pointed covers’ (Dress et al. 2012 , Theorem 7) and ‘stable triplet covers’ (Huber and Steel 2014 , Theorem 1)].…”

Section: Introductionmentioning

confidence: 99%

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Minimum triplet covers of binary phylogenetic X-trees

2017

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Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any ‘triplet cover’ of the tree (based on the fact that each non-leaf vertex is the median vertex of three leaves). In this paper we describe a number of new results concerning triplet covers of minimum size. In particular, we characterize such covers in terms of an associated graph being a 2-tree. Also, we show that minimum triplet covers are ‘shellable’ and thereby provide a set of pairs for which the inter-leaf distance values will uniquely determine the underlying tree and its associated branch lengths.

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Section: Introductionsupporting

confidence: 81%

Section: A Characterization Of Minimum Triplet Coversmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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Minimum triplet covers of binary phylogenetic X-trees

2017

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“…The triplet cover question posed in [3] and discussed further in [5], asks the following. Suppose that T is a triplet cover for a binary phylogenetic tree Xtree T having strictly positive edge lengths.…”

Section: Tree Distances From Any Triplet Cover Determines the Underlymentioning

confidence: 99%

Combinatorial properties of triplet covers for binary trees

Grünewald

Huber

Moulton

et al. 2018

Advances in Applied Mathematics

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It is a classical result that an unrooted tree T having positive real-valued edge lengths and no vertices of degree two can be reconstructed from the induced distance between each pair of leaves. Moreover, if each non-leaf vertex of T has degree 3 then the number of distance values required is linear in the number of leaves. A canonical candidate for such a set of pairs of leaves in T is the following:for each non-leaf vertex v, choose a leaf in each of the three components of T −v, group these three leaves into three pairs, and take the union of this set over all choices of v. This forms a so-called 'triplet cover' for T . In the first part of this paper we answer an open question (from 2012) by showing that the induced leaf-to-leaf distances for any triplet cover for T uniquely determine T and its edge lengths. We then investigate the finer combinatorial properties of triplet covers. In particular, we describe the structure of triplet covers that satisfy one or more of the following properties of being minimal, 'sparse', and 'shellable'.

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Lassoing and Corralling Rooted Phylogenetic Trees

Huber

Popescu

2013

Bull Math Biol

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The construction of a dendogram on a set of individuals is a key component of a genomewide association study. However, even with modern sequencing technologies the distances on the individuals required for the construction of such a structure may not always be reliable making it tempting to exclude them from an analysis. This, in turn, results in an input set for dendogram construction that consists of only partial distance information, which raises the following fundamental question. For what (proper) subsets of a dendogram's leaf set can we uniquely reconstruct the dendogram from the distances that it induces on the elements of such a subset? By formalizing a dendogram in terms of an edge-weighted, rooted, phylogenetic tree on a pre-given finite set X with |X|≥3 whose edge-weighting is equidistant and subsets Y of X for which the distances between every pair of elements in Y is known in terms of sets [Formula: see text] of 2-subsets of X, we investigate this problem from the perspective of when such a tree is lassoed, that is, uniquely determined by the elements in [Formula: see text]. For this, we consider four different formalizations of the idea of "uniquely determining" giving rise to four distinct types of lassos. We present characterizations for all of them in terms of the child-edge graphs of the interior vertices of such a tree. Our characterizations imply in particular that in case the tree in question is binary, then all four types of lasso must coincide.

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Tree Reconstruction from Triplet Cover Distances

Cited by 5 publications

References 10 publications

Minimum triplet covers of binary phylogenetic X-trees

Minimum triplet covers of binary phylogenetic X-trees

Combinatorial properties of triplet covers for binary trees

Lassoing and Corralling Rooted Phylogenetic Trees

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