On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems

Iersel, Leo van; Kelk, Steven; Stamoulis, Georgios; Stougie, Leen; Boes, Olivier

doi:10.1007/s00453-017-0366-5

Cited by 22 publications

(29 citation statements)

References 39 publications

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“…Unrooted minimum hybridization. In [29], it was shown that computing the TBR distance for a pair of unrooted binary phylogenetic trees T and T is equivalent to a problem that is concerned with computing the minimum number of extra edges required to simultaneously explain T and T . To describe this problem precisely, let N be an unrooted binary phylogenetic network on X, and let T be an unrooted binary phylogenetic tree on X.…”

Section: Preliminariesmentioning

confidence: 99%

“…where the minimum is taken over all unrooted binary phylogenetic networks on X that display T and T . The value h u (T, T ) is known as the hybridization number of T and T [29].…”

Section: Preliminariesmentioning

confidence: 99%

“…Many such kernelizations use subtree reductions and variations of chain reductions. The reductions typically have a common core but details differ from case to case depending on the specific combinatorial nature of the problem at hand: there are many subtle differences between TBR distance, SPR distance [4,6,32], hybridization number [7,29,30] and agreement forests [24,31], for example. Other relevant factors include whether the input trees are unrooted or rooted; whether the input trees are binary or non-binary and the number of trees allowed in the input (see earlier references and [28]).…”

Section: Rooted-hybridization-number (Rhn)mentioning

confidence: 99%

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A Tight Kernel for Computing the Tree Bisection and Reconnection Distance between Two Phylogenetic Trees

Kelk

Linz

2019

SIAM J. Discrete Math.

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In 2001 Allen and Steel showed that, if subtree and chain reduction rules have been applied to two unrooted phylogenetic trees, the reduced trees will have at most 28k taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees. Here we reanalyse Allen and Steel's kernelization algorithm and prove that the reduced instances will in fact have at most 15k − 9 taxa. Moreover we show, by describing a family of instances which have exactly 15k − 9 taxa after reduction, that this new bound is tight. These instances also have no common clusters, showing that a third commonly-encountered reduction rule, the cluster reduction, cannot further reduce the size of the kernel in the worst case. To achieve these results we introduce and use "unrooted generators" which are analogues of rooted structures that have appeared earlier in the phylogenetic networks literature. Using similar argumentation we show that, for the minimum hybridization problem on two rooted trees, 9k − 2 is a tight bound (when subtree and chain reduction rules have been applied) and 9k − 4 is a tight bound (when, additionally, the cluster reduction has been applied) on the number of taxa, where k is the hybridization number of the two trees.

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Section: Preliminariesmentioning

confidence: 99%

“…where the minimum is taken over all unrooted binary phylogenetic networks on X that display T and T . The value h u (T, T ) is known as the hybridization number of T and T [29].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Rooted-hybridization-number (Rhn)mentioning

confidence: 99%

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A Tight Kernel for Computing the Tree Bisection and Reconnection Distance between Two Phylogenetic Trees

Kelk

Linz

2019

SIAM J. Discrete Math.

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“…We emphasize that unrooted phylogenetic networks (as defined here and in e.g. [23,44,21,40]) should be viewed as undirected analogues of rooted phylogenetic networks, which correspond to directed graphs [29]. This is to distinguish them from split networks which are phylogenetic data-visualisation tools and which have a very different phylogenetic interpretation; these are sometimes also referred to as "unrooted" networks [36].Display graphs involving networks are relevant because of the growing number of optimization problems, traditionally posed on rooted trees and networks, which are now being mapped to the unrooted setting (see e.g.…”

mentioning

confidence: 99%

“…This is to distinguish them from split networks which are phylogenetic data-visualisation tools and which have a very different phylogenetic interpretation; these are sometimes also referred to as "unrooted" networks [36].Display graphs involving networks are relevant because of the growing number of optimization problems, traditionally posed on rooted trees and networks, which are now being mapped to the unrooted setting (see e.g. [31,44,27,21]). We prove that, if N displays T -i.e.…”

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confidence: 99%

Treewidth of display graphs: bounds, brambles and applications

Janssen¹,

Jones

Kelk³

et al. 2019

JGAA

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Phylogenetic trees and networks are leaf-labelled graphs used to model evolution. Display graphs are created by identifying common leaf labels in two or more phylogenetic trees or networks. The treewidth of such graphs is bounded as a function of many common dissimilarity measures between phylogenetic trees and this has been leveraged in fixed parameter tractability results. Here we further elucidate the properties of display graphs and their interaction with treewidth. We show that it is NP-hard to recognize display graphs, but that display graphs of bounded treewidth can be recognized in linear time. Next we show that if a phylogenetic network displays (i.e. topologically embeds) a phylogenetic tree, the treewidth of their display graph is bounded by a function of the treewidth of the original network (and also by various other parameters). In fact, using a bramble argument we show that this treewidth bound is sharp up to an additive term of 1. We leverage this bound to give an FPT algorithm, parameterized by treewidth, for determining whether a network displays a tree, which is an intensively-studied problem in the field. We conclude with a discussion on the future use of display graphs and treewidth in phylogenetics. arXiv:1809.00907v1 [cs.DS] 4 Sep 2018The purpose of this article is to further investigate, and algorithmically exploit, properties of the display graphs formed not only by trees, but also by trees and networks. To the best of our knowledge this is the first time tree-network display graphs have been considered. In the first part of the article, we list some basic properties of display graphs, and then address the problem of recognizing them, a problem posed in [33]. Specifically: given a cubic graph G, do there exist two unrooted binary phylogenetic trees T 1 , T 2 on the same set of taxa X such that G is the display graph D(T 1 , T 2 ) of T 1 and T 2 (after suppression of degree-2 nodes)? We prove that the problem is NP-hard, by providing an equivalence with the NP-hard TreeArboricity problem [13]. On the positive side, we prove that if G has bounded treewidth then this question can be answered in linear time. For this purpose we use Courcelle's Theorem [14,2]. This well-known meta-theorem states, essentially, that graph properties which can be expressed as a bounded-length fragment of Monadic Second Order Logic (MSOL) can be solved in linear time on graphs of bounded treewidth. We provide such an expression for recognizing display graphs.In the second, longer part of the article, we turn our attention to display graphs formed by merging an unrooted binary phylogenetic tree T with an unrooted binary phylogenetic network N , both on the same set of taxa X. The latter is simply an undirected graph where internal nodes have degree 3 and leaves, as usual, are bijectively labelled by X. Unlike trees, networks do not need to be acyclic. We emphasize that unrooted phylogenetic networks (as defined here and in e.g. [23,44,21,40]) should be viewed as undirected analogues of rooted phylogenetic networks,...

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New Reduction Rules for the Tree Bisection and Reconnection Distance

Kelk

Linz

2020

Ann. Comb.

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Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most $$15k-9$$ 15 k - 9 taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here, we propose five new reduction rules and show that these further reduce the bound to $$11k-9$$ 11 k - 9 . The new rules combine the “unrooted generator” approach introduced in Kelk and Linz (SIAM J Discrete Math 33(3):1556–1574, 2019) with a careful analysis of agreement forests to identify (i) situations when chains of length 3 can be further shortened without reducing the TBR distance, and (ii) situations when small subtrees can be identified whose deletion is guaranteed to reduce the TBR distance by 1. To the best of our knowledge these are the first reduction rules that strictly enhance the reductive power of the subtree and chain reduction rules.

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On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems

Cited by 22 publications

References 39 publications

A Tight Kernel for Computing the Tree Bisection and Reconnection Distance between Two Phylogenetic Trees

A Tight Kernel for Computing the Tree Bisection and Reconnection Distance between Two Phylogenetic Trees

Treewidth of display graphs: bounds, brambles and applications

New Reduction Rules for the Tree Bisection and Reconnection Distance

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