When Can Splits be Drawn in the Plane?

Balvočiūtė, Monika; Bryant, David; Spillner, Andreas

doi:10.1137/15m1040852

Cited by 3 publications

(9 citation statements)

References 24 publications

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“…To prove Proposition 1 we shall also use [21,Theorem 15] which states that a maximum linearly independent split system S on a set X with n ≥ 2 elements is a maximum flat split system if and only if, for every 4-element subset Y ⊆ X, the restriction S| Y contains precisely 6 splits.…”

Section: Discussionmentioning

confidence: 99%

“…Hence M = {a, b, c, d} ∪ {u, u , v, v } is a 6-element subset such that S D| M is not compatible. [21], the lemma can be proven using graph-theoretical concepts from [22]. In the following we provide a direct proof for completeness.…”

Section: Appendix Amentioning

confidence: 95%

“…Next assume that S is a maximum linearly independent split system that satisfies the pairwise separation property. Then, for n ∈ {2, 3, 4}, the fact that S consists of precisely n 2 splits immediately implies that S is a maximum flat split system by [21,Theorem 15].…”

Section: Proof Of Propositionmentioning

confidence: 99%

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Order Distances and Split Systems

Moulton

Spillner

2021

Order

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Given a pairwise distance D on the elements in a finite set X, the order distanceΔ(D) on X is defined by first associating a total preorder ≼x on X to each x ∈X based on D, and then quantifying the pairwise disagreement between these total preorders. The order distance can be useful in relational analyses because using Δ(D) instead of D may make such analyses less sensitive to small variations in D. Relatively little is known about properties of Δ(D) for general distances D. Indeed, nearly all previous work has focused on understanding the order distance of a treelike distance, that is, a distance that arises as the shortest path distances in a tree with non-negative edge weights and X mapped into its vertex set. In this paper we study the order distance Δ(D) for distances D that can be decomposed into sums of simpler distances called split-distances. Such distances D generalize treelike distances, and have applications in areas such as classification theory and phylogenetics.

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Section: Discussionmentioning

confidence: 99%

Section: Appendix Amentioning

confidence: 95%

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Order Distances and Split Systems

Moulton

Spillner

2021

Order

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“…X. Note that the splits in S D are precisely those appearing in the index set of the first sum in Equation (1). We chose the name for S D since it is closely related to the midpath phylogeny introduced in [20].…”

Section: The Midpath Split System Of a Distancementioning

confidence: 99%

“…Then, in Section 3, we prove a variant of Bonnot et al's result for arbitrary q ≥ p 2 > 0 in the special case where the split system underlying a tree is maximal (Theorem 1). In Section 4 we focus on the split system associated to a distance D on X that forms the index set for the first sum in Equation (1). In particular, we give a tight upper bound on its size (Theorem 2), and also a characterization for when it is compatible (Theorem 3).…”

Section: Introductionmentioning

confidence: 99%

Order distances and split systems

Moulton¹,

Spillner²

2019

Preprint

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Given a distance D on a finite set X with n elements, it is interesting to understand how the ranking Rx = z1, z2, . . . , zn obtained by ordering the elements in X according to increasing distance D(x, zi) from x, varies with different choices of x ∈ X. The order distance Op,q(D) is a distance on X associated to D which quantifies these variations, where q ≥ p 2 > 0 are parameters that control how ties in the rankings are handled. The order distance Op,q(D) of a distance D has been intensively studied in case D is a treelike distance (that is, D arises as the shortest path distances in an edge-weighted tree with leaves labeled by X), but relatively little is known about properties of Op,q(D) for general D. In this paper we study the order distance for various types of distances that naturally generalize treelike distances in that they can be generated by split systems, i.e. they are examples of so-called l1-distances. In particular we show how and to what extent properties of the split systems associated to the distances D that we study can be used to infer properties of Op,q(D).

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NeighborNet: improved algorithms and implementation

Bryant,

Huson

2023

Front. Bioinform.

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NeighborNet constructs phylogenetic networks to visualize distance data. It is a popular method used in a wide range of applications. While several studies have investigated its mathematical features, here we focus on computational aspects. The algorithm operates in three steps. We present a new simplified formulation of the first step, which aims at computing a circular ordering. We provide the first technical description of the second step, the estimation of split weights. We review the third step by constructing and drawing the network. Finally, we discuss how the networks might best be interpreted, review related approaches, and present some open questions.

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When Can Splits be Drawn in the Plane?

Cited by 3 publications

References 24 publications

Order Distances and Split Systems

Order Distances and Split Systems

Order distances and split systems

NeighborNet: improved algorithms and implementation

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