arXiv subjects

Katharina T. Huber

Publications and source records attributed to Katharina T. Huber.

At least 37 records · Page 2Linked to original sources

arXiv2021

The space of equidistant phylogenetic cactuses

We introduce and investigate the space of \emph{equidistant} $X$-\emph{cactuses}. These are rooted, arc weighted, phylogenetic networks with leaf set $X$, where $X$ is a finite set of species, and all leaves have the same distance from the root. The space contains as a subset the space of ultrametric trees on $X$ that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning \emph{ranked} rooted $X$-cactuses. In particular, we show that such networks can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of $X$ that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted $X$-trees in terms of partitions of $X$, which provides an alternative proof that the space of ultrametric trees on $X$ is CAT(0). As with spaces of phylogenetic trees, we expect that our results should provide the basis for and new directions in performing statistical analyses for collections of phylogenetic networks with arc lengths.

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arXiv2021

Encoding and ordering X-cactuses

Phylogenetic networks are a generalization of evolutionary or phylogenetic trees that are commonly used to represent the evolution of species which cross with one another. A special type of phylogenetic network is an {\em $X$-cactus}, which is essentially a cactus graph in which all vertices with degree less than three are labelled by at least one element from a set $X$ of species. In this paper, we present a way to {\em encode} $X$-cactuses in terms of certain collections of partitions of $X$ that naturally arise from $X$-cactuses. Using this encoding, we also introduce a partial order on the set of $X$-cactuses (up to isomorphism), and derive some structural properties of the resulting partially ordered set. This includes an analysis of some properties of its least upper and greatest lower bounds. Our results not only extend some fundamental properties of phylogenetic trees to $X$-cactuses, but also provides a new approach to solving topical problems in phylogenetic network theory such as deriving consensus networks.

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arXiv2021

Level-$2$ networks from shortest and longest distances

Recently it was shown that a certain class of phylogenetic networks, called level-$2$ networks, cannot be reconstructed from their associated distance matrices. In this paper, we show that they can be reconstructed from their induced shortest and longest distance matrices. That is, if two level-$2$ networks induce the same shortest and longest distance matrices, then they must be isomorphic. We further show that level-$2$ networks are reconstructible from their shortest distance matrices if and only if they do not contain a subgraph from a family of graphs. A generator of a network is the graph obtained by deleting all pendant subtrees and suppressing degree-$2$ vertices. We also show that networks with a leaf on every generator side is reconstructible from their induced shortest distance matrix, regardless of level.

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arXiv2021

Phylogenetic consensus networks: Computing a consensus of 1-nested phylogenetic networks

An important and well-studied problem in phylogenetics is to compute a \emph{consensus tree} so as to summarize the common features within a collection of rooted phylogenetic trees, all whose leaf-sets are bijectively labeled by the same set~(X) of species. More recently, however, it has become of interest to find a consensus for a collection of more general, rooted directed acyclic graphs all of whose sink-sets are bijectively labeled by~(X), so called rooted \emph{phylogenetic networks}. These networks are used to analyse the evolution of species that cross with one another, such as plants and viruses. In this paper, we introduce an algorithm for computing a consensus for a collection of so-called 1-\emph{nested} phylogenetic networks. Our approach builds on a previous result by Roselló et al. that describes an encoding for any 1-nested phylogenetic network in terms of a collection of ordered pairs of subsets of (X).More specifically, we characterize those collections of ordered pairs that arise as the encoding of some 1-nested phylogenetic network, and then use this characterization to compute a \emph{consensus network} for a collection of~$t$ 1-nested networks in $O(t|X|^2+|X|^3)$ time. Applying our algorithm to a collection of phylogenetic trees yields the well-known majority rule consensus tree. Our approach leads to several new directions for futurework, and we expect that it should provide a useful new tool to help understand complex evolutionary scenarios.

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arXiv2020

Weakly displaying trees in temporal tree-child network

Recently there has been considerable interest in the problem of finding a phylogenetic network with a minimum number of reticulation vertices which displays a given set of phylogenetic trees, that is, a network with minimum hybrid number. Even so, for certain evolutionary scenarios insisting that a network displays the set of trees can be an overly restrictive assumption. In this paper, we consider the less restrictive notion of displaying called weakly displaying and, in particular, a special case of this which we call rigidly displaying. We characterize when two trees can be rigidly displayed by a temporal tree-child network in terms of fork-picking sequences, a concept that is closely related to that of cherry-picking sequences. We also show that, in case it exists, the rigid hybrid number for two phylogenetic trees is given by a minimum weight fork-picking sequence for the trees, and that the rigid hybrid number can be quite different from the related beaded- and temporal-hybrid numbers.

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arXiv2020

Recognizing and realizing cactus metrics

The problem of realizing finite metric spaces in terms of weighted graphs has many applications. For example, the mathematical and computational properties of metrics that can be realized by trees have been well-studied and such research has laid the foundation of the reconstruction of phylogenetic trees from evolutionary distances. However, as trees may be too restrictive to accurately represent real-world data or phenomena, it is important to understand the relationship between more general graphs and distances. In this paper, we introduce a new type of metric called a cactus metric, that is, a metric that can be realized by a cactus graph. We show that, just as with tree metrics, a cactus metric has a unique optimal realization. In addition, we describe an algorithm that can recognize whether or not a metric is a cactus metric and, if so, compute its optimal realization in $O(n^3)$ time, where $n$ is the number of points in the space.

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arXiv2019

Phylogenetic networks that are their own fold-ups

Phylogenetic networks are becoming of increasing interest to evolutionary biologists due to their ability to capture complex non-treelike evolutionary processes. From a combinatorial point of view, such networks are certain types of rooted directed acyclic graphs whose leaves are labelled by, for example, species. A number of mathematically interesting classes of phylogenetic networks are known. These include the biologically relevant class of stable phylogenetic networks whose members are defined via certain "fold-up" and "un-fold" operations that link them with concepts arising within the theory of, for example, graph fibrations. Despite this exciting link, the structural complexity of stable phylogenetic networks is still relatively poorly understood. Employing the popular tree-based, reticulation-visible, and tree-child properties which allow one to gauge this complexity in one way or another, we provide novel characterizations for when a stable phylogenetic network satisfies either one of these three properties.

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arXiv2019

Reconciling Event-Labeled Gene Trees with MUL-trees and Species Networks

Phylogenomics commonly aims to construct evolutionary trees from genomic sequence information. One way to approach this problem is to first estimate event-labeled gene trees (i.e., rooted trees whose non-leaf vertices are labeled by speciation or gene duplication events), and to then look for a species tree which can be reconciled with this tree through a \emph{reconciliation map} between the trees. In practice, however, it can happen that there is no such map from a given event-labeled tree to \emph{any} species tree. An important situation where this might arise is where the species evolution is better represented by a \emph{network} instead of a tree. In this paper, we therefore consider the problem of reconciling event-labeled trees with species networks. In particular, we prove that any event-labeled gene tree can be reconciled with some network and that, under certain mild assumptions on the gene tree, the network can even be assumed to be multi-arc free. To prove this result, we show that we can always reconcile the gene tree with some multi-labeled (MUL-)tree, which can then be "folded up" to produce the desired reconciliation and network. In addition, we study the interplay between reconciliation maps from event-labeled gene trees to MUL-trees and networks. Our results could be useful for understanding how genomes have evolved after undergoing complex evolutionary events such as polyploidy.

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arXiv2018

The complexity of comparing multiply-labelled trees by extending phylogenetic-tree metrics

A multilabeled tree (or MUL-tree) is a rooted tree in which every leaf is labelled by an element from some set, but in which more than one leaf may be labelled by the same element of that set. In phylogenetics, such trees are used in biogeographical studies, to study the evolution of gene families, and also within approaches to construct phylogenetic networks. A multilabelled tree in which no leaf-labels are repeated is called a phylogenetic tree, and one in which every label is the same is also known as a tree-shape. In this paper, we consider the complexity of computing metrics on MUL-trees that are obtained by extending metrics on phylogenetic trees. In particular, by restricting our attention to tree shapes, we show that computing the metric extension on MUL-trees is NP complete for two well-known metrics on phylogenetic trees, namely, the path-difference and Robinson Foulds distances. We also show that the extension of the Robinson Foulds distance is fixed parameter tractable with respect to the distance parameter. The path distance complexity result allows us to also answer an open problem concerning the complexity of solving the quadratic assignment problem for two matrices that are a Robinson similarity and a Robinson dissimilarity, which we show to be NP-complete. We conclude by considering the maximum agreement subtree (MAST) distance on phylogenetic trees to MUL-trees. Although its extension to MUL-trees can be computed in polynomial time, we show that computing its natural generalization to more than two MUL-trees is NP-complete, although fixed-parameter tractable in the maximum degree when the number of given trees is bounded.

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arXiv2018

Phylogenetic flexibility via Hall-type inequalities and submodularity

Given a collection $τ$ of subsets of a finite set $X$, we say that $τ$ is {\em phylogenetically flexible} if, for any collection $R$ of rooted phylogenetic trees whose leaf sets comprise the collection $τ$, $R$ is compatible (i.e. there is a rooted phylogenetic $X$--tree that displays each tree in $R$). We show that $τ$ is phylogenetically flexible if and only if it satisfies a Hall-type inequality condition of being `slim'. Using submodularity arguments, we show that there is a polynomial-time algorithm for determining whether or not $τ$ is slim. This `slim' condition reduces to a simpler inequality in the case where all of the sets in $τ$ have size 3, a property we call `thin'. Thin sets were recently shown to be equivalent to the existence of an (unrooted) tree for which the median function provides an injective mapping to its vertex set; we show here that the unrooted tree in this representation can always be chosen to be a caterpillar tree. We also characterise when a collection $τ$ of subsets of size 2 is thin (in terms of the flexibility of total orders rather than phylogenies) and show that this holds if and only if an associated bipartite graph is a forest. The significance of our results for phylogenetics is in providing precise and efficiently verifiable conditions under which supertree methods that require consistent inputs of trees, can be applied to any input trees on given subsets of species.

math.CO↗

arXiv2017

Three-way symbolic tree-maps and ultrametrics

Three-way dissimilarities are a generalization of (two-way) dissimilarities which can be used to indicate the lack of homogeneity or resemblance between any three objects. Such maps have applications in cluster analysis, and have been used in areas such as psychology and phylogenetics, where three-way data tables can arise. Special examples of such dissimilarities are three-way tree-metrics and ultrametrics, which arise from leaf-labelled trees with edges labelled by positive real numbers. Here we consider three-way maps which arise from leaf-labelled trees where instead the interior vertices are labelled by an arbitrary set of values. For unrooted trees we call such maps three-way symbolic tree-maps; for rooted trees we call them three-way symbolic ultrametrics since they can be considered as a generalization of the (two-way) symbolic ultrametrics of Böcker and Dress. We show that, as with two- and three-way tree-metrics and ultrametrics, three-way symbolic tree-maps and ultrametrics can be characterized via certain $k$-point conditions. In the unrooted case, our characterization is mathematically equivalent to one presented by Gurvich for a certain class of edge-labelled hypergraphs. We also show that it can be decided whether or not an arbitrary three-way symbolic map is a tree-map or a symbolic ultrametric using a triplet-based approach that relies on the so-called BUILD algorithm for deciding when a set of 3-leaved trees or triplets can be displayed by a single tree. We envisage that our results will be useful in developing new approaches and algorithms for understanding 3-way data, especially within the area of phylogenetics.

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arXiv2017

Combinatorial properties of triplet covers for binary trees

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'.

math.CO↗

arXiv2017

Geometric medians in reconciliation spaces

In evolutionary biology, it is common to study how various entities evolve together, for example, how parasites coevolve with their host, or genes with their species. Coevolution is commonly modelled by considering certain maps or reconciliations from one evolutionary tree $P$ to another $H$, all of which induce the same map $ϕ$ between the leaf-sets of $P$ and $H$ (corresponding to present-day associations). Recently, there has been much interest in studying spaces of reconciliations, which arise by defining some metric $d$ on the set $Rec(P,H,ϕ)$ of all possible reconciliations between $P$ and $H$. In this paper, we study the following question: How do we compute a geometric median for a given subset $Ψ$ of $Rec(P,H,ϕ)$ relative to $d$, i.e. an element $ψ_{med} \in Rec(P,H,ϕ)$ such that $$ \sum_{ψ' \in Ψ} d(ψ_{med},ψ') \le \sum_{ψ' \in Ψ} d(ψ,ψ') $$ holds for all $ψ\in Rec(P,H,ϕ)$? For a model where so-called host-switches or transfers are not allowed, and for a commonly used metric $d$ called the edit-distance, we show that although the cardinality of $Rec(P,H,ϕ)$ can be super-exponential, it is still possible to compute a geometric median for a set $Ψ$ in $Rec(P,H,ϕ)$ in polynomial time. We expect that this result could be useful for computing a summary or consensus for a set of reconciliations (e.g. for a set of suboptimal reconciliations).

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arXiv2017

From event labeled gene trees to species trees

Background: Tree reconciliation problems have long been studied in phylogenetics. A particular variant of the reconciliation problem for a gene tree T and a species tree S assumes that for each interior vertex x of T it is known whether x represents a speciation or a duplication. This problem appears in the context of analyzing orthology data. Results: We show that S is a species tree for T if and only if S displays all rooted triples of T that have three distinct species as their leaves and are rooted in a speciation vertex. A valid reconciliation map can then be found in polynomial time. Simulated data shows that the event-labeled gene trees convey a large amount of information on underlying species trees, even for a large percentage of losses. Conclusions: The knowledge of event labels in a gene tree strongly constrains the possible species tree and, for a given species tree, also the possible reconciliation maps. Nevertheless, many degrees of freedom remain in the space of feasible solutions. In order to disambiguate the alternative solutions additional external constraints as well as optimization criteria could be employed.

cs.DM↗

arXiv2017

Minimum triplet covers of binary phylogenetic $X$-trees

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.

math.CO↗

arXiv2016

Transforming phylogenetic networks: Moving beyond tree space

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent reticulate evolution. Unrooted phylogenetic networks form a special class of such networks, which naturally generalize unrooted phylogenetic trees. In this paper we define two operations on unrooted phylogenetic networks, one of which is a generalization of the well-known nearest-neighbor interchange (NNI) operation on phylogenetic trees. We show that any unrooted phylogenetic network can be transformed into any other such network using only these operations. This generalizes the well-known fact that any phylogenetic tree can be transformed into any other such tree using only NNI operations. It also allows us to define a generalization of tree space and to define some new metrics on unrooted phylogenetic networks. To prove our main results, we employ some fascinating new connections between phylogenetic networks and cubic graphs that we have recently discovered. Our results should be useful in developing new strategies to search for optimal phylogenetic networks, a topic that has recently generated some interest in the literature, as well as for providing new ways to compare networks.

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arXiv2015

Bridging the gap between rooted and unrooted phylogenetic networks

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard phylogenetic tree model by also allowing cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees, however, surprisingly little is known about the interplay between both types thus hampering our ability to make much needed progress for rooted phylogenetic networks by drawing on insights from their much better understood unrooted counterparts. Unrooted phylogenetic networks are underpinned by split systems and by focusing on them we establish a first link between both types. More precisely, we develop a link between 1-nested phylogenetic networks which are examples of rooted phylogenetic networks and the well-studied median networks (aka Buneman graph) which are examples of unrooted phylogenetic networks. In particular, we show that not only can a 1-nested network be obtained from a median network but also that that network is, in a well-defined sense, optimal. Along the way, we characterize circular split systems in terms of the novel $\mathcal I$-intersection closure of a split system and establish the 1-nested analogue of the fundamental "Splits Equivalence Theorem" for phylogenetic trees.

math.CO↗

arXiv2015

Folding and unfolding phylogenetic trees and networks

Phylogenetic networks are rooted, labelled directed acyclic graphs which are commonly used to represent reticulate evolution. There is a close relationship between phylogenetic networks and multi-labelled trees (MUL-trees). Indeed, any phylogenetic network $N$ can be 'unfolded' to obtain a MUL-tree $U(N)$ and, conversely, a MUL-tree $T$ can in certain circumstances be 'folded' to obtain a phylogenetic network $F(T)$ that exhibits $T$. In this paper, we study properties of the operations $U$ and $F$ in more detail. In particular, we introduce the class of stable networks, phylogenetic networks $N$ for which $F(U(N))$ is isomorphic to $N$, characterise such networks, and show that that they are related to the well-known class of tree-sibling networks. We also explore how the concept of displaying a tree in a network $N$ can be related to displaying the tree in the MUL-tree $U(N)$. To do this, we develop a phylogenetic analogue of graph fibrations. This allows us to view $U(N)$ as the analogue of the universal cover of a digraph, and to establish a close connection between displaying trees in $U(N)$ and reconciling phylogenetic trees with networks.

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