arXiv subjects

Vincent Moulton

Publications and source records attributed to Vincent Moulton.

At least 37 records · Page 2Linked to original sources

arXiv2023

Orienting undirected phylogenetic networks

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network $N$ can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of $N$ and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

cs.DS↗

arXiv2023

Is this network proper forest-based?

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether or not a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it can be decided in polynomial time whether or not a binary, tree-child network with $m \ge 2$ roots is proper forest-based in case $m=2$, but that this problem is NP-complete for $m\ge 3$. We also give a fixed parameter tractable (FPT) algorithm for deciding whether or not a network in which every vertex has indegree at most 2 is proper forest-based. A key element in proving our results is a new characterization for when a network with $m$ roots is proper forest-based which is given in terms of the existence of certain $m$-colorings of the vertices of the network.

q-bio.PE↗

arXiv2023

Shared ancestry graphs and symbolic arboreal maps

A network $N$ on a finite set $X$, $|X|\geq 2$, is a connected directed acyclic graph with leaf set $X$ in which every root in $N$ has outdegree at least 2 and no vertex in $N$ has indegree and outdegree equal to 1; $N$ is arboreal if the underlying unrooted, undirected graph of $N$ is a tree. Networks are of interest in evolutionary biology since they are used, for example, to represent the evolutionary history of a set $X$ of species whose ancestors have exchanged genes in the past. For $M$ some arbitrary set of symbols, $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if there exists some arboreal network $N$ whose vertices with outdegree two or more are labelled by elements in $M$ and so that $d(\{x,y\})$, $\{x,y\} \in {X \choose 2}$, is equal to the label of the least common ancestor of $x$ and $y$ in $N$ if this exists and $\odot$ else. Important examples of symbolic arboreal maps include the symbolic ultrametrics, which arise in areas such as game theory, phylogenetics and cograph theory. In this paper we show that a map $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if and only if $d$ satisfies certain 3- and 4-point conditions and the graph with vertex set $X$ and edge set consisting of those pairs $\{x,y\} \in {X \choose 2}$ with $d(\{x,y\}) \neq \odot$ is Ptolemaic. To do this, we introduce and prove a key theorem concerning the shared ancestry graph for a network $N$ on $X$, where this is the graph with vertex set $X$ and edge set consisting of those $\{x,y\} \in {X \choose 2}$ such that $x$ and $y$ share a common ancestor in $N$. In particular, we show that for any connected graph $G$ with vertex set $X$ and edge clique cover $K$ in which there are no two distinct sets in $K$ with one a subset of the other, there is some network with $|K|$ roots and leaf set $X$ whose shared ancestry graph is $G$.

math.CO↗

arXiv2023

Polynomial invariants for cactuses

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial $F$ that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial $Q$ that is a complete invariant for the class of rooted cactuses that has 6 variables \vince{whose degree can be bounded linearly in terms of the size of the rooted cactus}. We also explain how to extend the $Q$ polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.

math.CO↗

arXiv2023

Diversities and the Generalized Circumradius

The generalized circumradius of a set of points $A \subseteq \mathbb{R}^d$ with respect to a convex body $K$ equals the minimum value of $λ\geq 0$ such that $A$ is contained in a translate of $λK$. Each choice of $K$ gives a different function on the set of bounded subsets of $\mathbb{R}^d$; we characterize which functions can arise in this way. Our characterization draws on the theory of diversities, a recently introduced generalization of metrics from functions on pairs to functions on finite subsets. We additionally investigate functions which arise by restricting the generalised circumradius to a finite subset of $\mathbb{R}^d$. We obtain elegant characterizations in the case that $K$ is a simplex or parallelotope.

math.MG↗

arXiv2022

Planar Rooted Phylogenetic Networks

A rooted phylogenetic network is a directed acyclic graph with a single root, whose sinks correspond to a set of species. As such networks are useful for representing the evolution of species that have undergone reticulate evolution, there has been great interest in developing the theory behind and algorithms for constructing them. However, unlike evolutionary trees, these networks can be highly non-planar, which can make them difficult to visualise and interpret. Here we investigate properties of planar rooted phylogenetic networks and algorithms for deciding whether or not rooted networks have certain special planarity properties. In particular, we introduce three natural subclasses of planar rooted phylogenetic networks and show that they form a hierarchy. In addition, for the well-known level-k networks, we show that level-1, -2, -3 networks are always outer, terminal, and upward planar, respectively, and that level-4 networks are not necessarily planar. Finally, we show that a regular network is terminal planar if and only if it is pyramidal. Our results make use of the highly developed field of planar digraphs, and we believe that the link between phylogenetic networks and planar graphs should prove useful in future for developing new approaches to both construct and visualise phylogenetic networks.

q-bio.PE↗

arXiv2022

Posets and spaces of $k$-noncrossing RNA Structures

RNA molecules are single-stranded analogues of DNA that can fold into various structures which influence their biological function within the cell. RNA structures can be modelled combinatorially in terms of a certain type of graph called an RNA diagram. In this paper we introduce a new poset of RNA diagrams $\mathcal{B}^r_{f,k}$, $r\ge 0$, $k \ge 1$ and $f \ge 3$, which we call the Penner-Waterman poset, and, using results from the theory of multitriangulations, we show that this is a pure poset of rank $k(2f-2k+1)+r-f-1$, whose geometric realization is the join of a simplicial sphere of dimension $k(f-2k)-1$ and an $\left((f+1)(k-1)-1\right)$-simplex in case $r=0$. As a corollary for the special case $k=1$, we obtain a result due to Penner and Waterman concerning the topology of the space of RNA secondary structures. These results could eventually lead to new ways to investigate landscapes of RNA $k$-noncrossing structures.

math.CO↗

arXiv2022

Compatible split systems on a multiset

A split system on a multiset $\mathcal M$ is a set of bipartitions of $\mathcal M$. Such a split system $\mathfrak S$ is compatible if it can be represented by a tree in such a way that the vertices of the tree are labelled by the elements in $\mathcal M$, the removal of each edge in the tree yields a bipartition in $\mathfrak S$ by taking the labels of the two resulting components, and every bipartition in $\mathfrak S$ can be obtained from the tree in this way. In this contribution, we present a novel characterization for compatible split systems, and for split systems admitting a unique tree representation. In addition, we show that a conjecture on compatibility stated in 2008 holds for some large classes of split systems.

math.CO↗

arXiv2022

Forest-based networks

In evolutionary studies it is common to use phylogenetic trees to represent the evolutionary history of a set of species. However, in case the transfer of genes or other genetic information between the species or their ancestors has occurred in the past, a tree may not provide a complete picture of their history. In such cases,tree-based phylogenetic networks can provide a useful, more refined representation of the species evolution. Such a network is essentially a phylogenetic tree with some arcs added between the tree edges so as to represent reticulate events such as gene transfer. Even so, this model does not permit the representation of evolutionary scenarios where reticulate events have taken place between different subfamilies or lineages of species. To represent such scenarios, in this paper we introduce the notion of a forest-based phylogenetic network, that is, a collection of leaf-disjoint phylogenetic trees on a set of species with arcs added between the edges of distinct trees within the collection. Forest-based networks include the recently introduced class of overlaid species forests which are used to model introgression. As we shall see, even though the definition of forest-based networks is closely related to that of tree-based networks, they lead to new mathematical theory which complements that of tree-based networks. As well as studying the relationship of forest-based networks with other classes of phylogenetic networks, such as tree-child networks and universal tree-based networks, we present some characterizations of some special classes of forest-based networks. We expect that our results will be useful for developing new models and algorithms to understand reticulate evolution, such as gene transfer between collections of bacteria that live in different environments.

math.CO↗

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.

q-bio.PE↗

arXiv2021

Planar Median Graphs and Cubesquare-Graphs

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of ``square-boundary'' amalgamations. These considerations also lead to an $\mathcal{O}(n\log n)$-time recognition algorithm to compute a decomposition of a planar median graph with $n$ vertices into cubes and square-graphs.

math.CO↗

arXiv2021

An algorithm for reconstructing level-2 phylogenetic networks from trinets

Evolutionary histories for species that cross with one another or exchange genetic material can be represented by leaf-labelled, directed graphs called phylogenetic networks. A major challenge in the burgeoning area of phylogenetic networks is to develop algorithms for building such networks by amalgamating small networks into a single large network. The level of a phylogenetic network is a measure of its deviation from being a tree; the higher the level of network, the less treelike it becomes. Various algorithms have been developed for building level-1 networks from small networks. However, level-1 networks may not be able to capture the complexity of some data sets. In this paper, we present a polynomial-time algorithm for constructing a rooted binary level-2 phylogenetic network from a collection of 3-leaf networks or trinets. Moreover, we prove that the algorithm will correctly reconstruct such a network if it is given all of the trinets in the network as input. The algorithm runs in time $O(t\cdot n+n^4)$ with $t$ the number of input trinets and $n$ the number of leaves. We also show that there is a fundamental obstruction to constructing level-3 networks from trinets, and so new approaches will need to be developed for constructing level-3 and higher level-networks.

cs.DS↗

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.

q-bio.PE↗

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.

math.CO↗

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.

q-bio.PE↗

arXiv2020

Reconstructibility of unrooted level-$k$ phylogenetic networks from distances

A phylogenetic network is a graph-theoretical tool that is used by biologists to represent the evolutionary history of a collection of species. One potential way of constructing such networks is via a distance-based approach, where one is asked to find a phylogenetic network that in some way represents a given distance matrix, which gives information on the evolutionary distances between present-day taxa. Here, we consider the following question. For which~$k$ are unrooted level-$k$ networks uniquely determined by their distance matrices? We consider this question for shortest distances as well as for the case that the multisets of all distances is given. We prove that level-$1$ networks and level-$2$ networks are reconstructible from their shortest distances and multisets of distances, respectively. Furthermore we show that, in general, networks of level higher than~$1$ are not reconstructible from shortest distances and that networks of level higher than~$2$ are not reconstructible from their multisets of distances.

math.CO↗

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.

cs.DM↗

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.

math.CO↗