arXiv subjects

Charles Semple

Publications and source records attributed to Charles Semple.

At least 37 records · Page 2Linked to original sources

arXiv2022

Cyclic matroids

For all positive integers $s$ and $t$ exceeding one, a matroid $M$ on $n$ elements is {\em nearly $(s, t)$-cyclic} if there is a cyclic ordering $σ$ of its ground set such that every $s-1$ consecutive elements of $σ$ are contained in an $s$-element circuit and every $t-1$ consecutive elements of $σ$ are contained in a $t$-element cocircuit. In the case $s=t$, nearly $(s, s)$-cyclic matroids have been studied previously. In this paper, we show that if $M$ is nearly $(s, t)$-cyclic and $n$ is sufficiently large, then these $s$-element circuits and $t$-element cocircuits are consecutive in $σ$ in a prescribed way, that is, $M$ is "$(s, t)$-cyclic". Furthermore, we show that, given $s$ and $t$ where $t\ge s$, every $(s, t)$-cyclic matroid on $n > s+t-2$ elements is a weak-map image of the $\left(\frac{t-s}{2}\right)$-th truncation of a certain $(s, s)$-cyclic matroid. If $s=3$, this certain matroid is the rank-$\frac{n}{2}$ whirl, and if $s=4$, this certain matroid is the rank-$\frac{n}{2}$ free swirl.

math.CO↗

arXiv2022

Counting and optimising maximum phylogenetic diversity sets

In conservation biology, phylogenetic diversity (PD) provides a way to quantify the impact of the current rapid extinction of species on the evolutionary `Tree of Life'. This approach recognises that extinction not only removes species but also the branches of the tree on which unique features shared by the extinct species arose. In this paper, we investigate three questions that are relevant to PD. The first asks how many sets of species of given size $k$ preserve the maximum possible amount of PD in a given tree. The number of such maximum PD sets can be very large, even for moderate-sized phylogenies. We provide a combinatorial characterisation of maximum PD sets, focusing on the setting where the branch lengths are ultrametric (e.g. proportional to time). This leads to a polynomial-time algorithm for calculating the number of maximum PD sets of size $k$ by applying a generating function; we also investigate the types of tree shapes that harbour the most (or fewest) maximum PD sets of size $k$. Our second question concerns optimising a linear function on the species (regarded as leaves of the phylogenetic tree) across all the maximum PD sets of a given size. Using the characterisation result from the first question, we show how this optimisation problem can be solved in polynomial time, even though the number of maximum PD sets can grow exponentially. Our third question considers a dual problem: If $k$ species were to become extinct, then what is the largest possible {\em loss} of PD in the resulting tree? For this question, we describe a polynomial-time solution based on dynamical programming.

q-bio.PE↗

arXiv2021

Elastic elements in 3-connected matroids

It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either ${\rm co}(M\delete e)$, the cosimplification of $M\delete e$, or ${\rm si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both ${\rm co}(M\delete e)$ and ${\rm si}(M/e)$ are $3$-connected. Calling such an element "elastic", in this paper we show that if $|E(M)|\ge 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans and, up to duality, $M$ has no $3$-separating set $S$ that is the disjoint union of a rank-$2$ subset and a corank-$2$ subset of $E(M)$ such that $M|S$ is isomorphic to a member or a single-element deletion of a member of a certain family of matroids.

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arXiv2021

On the Complexity of Optimising Variants of Phylogenetic Diversity on Phylogenetic Networks

Phylogenetic Diversity (PD) is a prominent quantitative measure of the biodiversity of a collection of present-day species (taxa). This measure is based on the evolutionary distance among the species in the collection. Loosely speaking, if $\mathcal{T}$ is a rooted phylogenetic tree whose leaf set $X$ represents a set of species and whose edges have real-valued lengths (weights), then the PD score of a subset $S$ of $X$ is the sum of the weights of the edges of the minimal subtree of $\mathcal{T}$ connecting the species in $S$. In this paper, we define several natural variants of the PD score for a subset of taxa which are related by a known rooted phylogenetic network. Under these variants, we explore, for a positive integer $k$, the computational complexity of determining the maximum PD score over all subsets of taxa of size $k$ when the input is restricted to different classes of rooted phylogenetic networks

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arXiv2021

Non-essential arcs in phylogenetic networks

In the study of rooted phylogenetic networks, analyzing the set of rooted phylogenetic trees that are embedded in such a network is a recurring task. From an algorithmic viewpoint, this analysis almost always requires an exhaustive search of a particular multiset $S$ of rooted phylogenetic trees that are embedded in a rooted phylogenetic network $\mathcal{N}$. Since the size of $S$ is exponential in the number of reticulations of $\mathcal{N}$, it is consequently of interest to keep this number as small as possible but without loosing any element of $S$. In this paper, we take a first step towards this goal by introducing the notion of a non-essential arc of $\mathcal{N}$, which is an arc whose deletion from $\mathcal{N}$ results in a rooted phylogenetic network $\mathcal{N}'$ such that the sets of rooted phylogenetic trees that are embedded in $\mathcal{N}$ and $\mathcal{N}'$ are the same. We investigate the popular class of tree-child networks and characterize which arcs are non-essential. This characterization is based on a family of directed graphs. Using this novel characterization, we show that identifying and deleting all non-essential arcs in a tree-child network takes time that is cubic in the number of leaves of the network. Moreover, we show that deciding if a given arc of an arbitrary phylogenetic network is non-essential is $Π_2^P$-complete.

math.CO↗

arXiv2020

Trinets encode orchard phylogenetic networks

Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if $\mathcal T$ and $\mathcal T'$ are rooted binary phylogenetic $X$-trees that infers the same set of rooted triples, then $\mathcal T$ and $\mathcal T'$ are isomorphic. However, in general, this sufficiency does not extend to rooted binary phylogenetic networks. In this paper, we show that trinets, phylogenetic network analogues of rooted triples, are sufficient to encode rooted binary orchard networks. Rooted binary orchard networks naturally generalise rooted binary tree-child networks. Moreover, we present a polynomial-time algorithm for building a rooted binary orchard network from its set of trinets. As a consequence, this algorithm affirmatively answers a previously-posed question of whether there is a polynomial-time algorithm for building a rooted binary tree-child network from the set of trinets it infers.

math.CO↗

arXiv2020

Defining phylogenetic networks using ancestral profiles

Rooted phylogenetic networks provide a more complete representation of the ancestral relationship between species than phylogenetic trees when reticulate evolutionary processes are at play. One way to reconstruct a phylogenetic network is to consider its `ancestral profile' (the number of paths from each ancestral vertex to each leaf). In general, this information does not uniquely determine the underlying phylogenetic network. A recent paper considered a new class of phylogenetic networks called `orchard networks' where this uniqueness was claimed to hold. Here we show that an additional restriction on the network, that of being `stack-free', is required in order for the original uniqueness claim to hold. On the other hand, if the additional stack-free restriction is lifted, we establish an alternative result; namely, there is uniqueness within the class of orchard networks up to the resolution of vertices of high in-degree.

math.CO↗

arXiv2020

On the maximum agreement subtree conjecture for balanced trees

We give a counterexample to the conjecture of Martin and Thatte that two balanced rooted binary leaf-labelled trees on $n$ leaves have a maximum agreement subtree (MAST) of size at least $n^{\frac{1}{2}}$. In particular, we show that for any $c>0$, there exist two balanced rooted binary leaf-labelled trees on $n$ leaves such that any MAST for these two trees has size less than $c n^{\frac{1}{2}}$. We also improve the lower bound of the size of such a MAST to $n^{\frac{1}{6}}$.

math.CO↗

arXiv2020

Caterpillars on three and four leaves are sufficient to reconstruct normal networks

While every rooted binary phylogenetic tree is determined by its set of displayed rooted triples, such a result does not hold for an arbitrary rooted binary phylogenetic network. In particular, there exist two non-isomorphic rooted binary temporal normal networks that display the same set of rooted triples. Moreover, without any structural constraint on the rooted phylogenetic networks under consideration, similarly negative results have also been established for binets and trinets which are rooted subnetworks on two and three leaves, respectively. Hence, in general, piecing together a rooted phylogenetic network from such a set of small building blocks appears insurmountable. In contrast to these results, in this paper, we show that a rooted binary normal network is determined by its sets of displayed caterpillars (particular type of subtrees) on three and four leaves. The proof is constructive and realises a polynomial-time algorithm that takes the sets of caterpillars on three and four leaves displayed by a rooted binary normal network and, up to isomorphism, reconstructs this network.

math.CO↗

arXiv2019

Placing quantified variants of 3-SAT and Not-All-Equal 3-SAT in the polynomial hierarchy

The complexity of variants of 3-SAT and Not-All-Equal 3-SAT is well studied. However, in contrast, very little is known about the complexity of the problems' quantified counterparts. In the first part of this paper, we show that $\forall \exists$ 3-SAT is $Π_2^P$-complete even if (1) each variable appears exactly twice unnegated and exactly twice negated, (2) each clause is a disjunction of exactly three distinct variables, and (3) the number of universal variables is equal to the number of existential variables. Furthermore, we show that the problem remains $Π_2^P$-complete if (1a) each universal variable appears exactly once unnegated and exactly once negated, (1b) each existential variable appears exactly twice unnegated and exactly twice negated, and (2) and (3) remain unchanged. On the other hand, the problem becomes NP-complete for certain variants in which each universal variable appears exactly once. In the second part of the paper, we establish $Π_2^P$-completeness for $\forall \exists$ Not-All-Equal 3-SAT even if (1') the Boolean formula is linear and monotone, (2') each universal variable appears exactly once and each existential variable appears exactly three times, and (3') each clause is a disjunction of exactly three distinct variables that contains at most one universal variable. On the positive side, we uncover variants of $\forall \exists$ Not-All-Equal 3-SAT that are co-NP-complete or solvable in polynomial time.

cs.CC↗

arXiv2019

A class of phylogenetic networks reconstructable from ancestral profiles

Rooted phylogenetic networks provide an explicit representation of the evolutionary history of a set $X$ of sampled species. In contrast to phylogenetic trees which show only speciation events, networks can also accommodate reticulate processes (for example, hybrid evolution, endosymbiosis, and lateral gene transfer). A major goal in systematic biology is to infer evolutionary relationships, and while phylogenetic trees can be uniquely determined from various simple combinatorial data on $X$, for networks the reconstruction question is much more subtle. Here we ask when can a network be uniquely reconstructed from its `ancestral profile' (the number of paths from each ancestral vertex to each element in $X$). We show that reconstruction holds (even within the class of all networks) for a class of networks we call `orchard networks', and we provide a polynomial-time algorithm for reconstructing any orchard network from its ancestral profile. Our approach relies on establishing a structural theorem for orchard networks, which also provides for a fast (polynomial-time) algorithm to test if any given network is of orchard type. Since the class of orchard networks includes tree-sibling tree-consistent networks and tree-child networks, our result generalise reconstruction results from 2008 and 2009. Orchard networks allow for an unbounded number $k$ of reticulation vertices, in contrast to tree-sibling tree-consistent networks and tree-child networks for which $k$ is at most $2|X|-4$ and $|X|-1$, respectively.

math.CO↗

arXiv2019

Displaying trees across two phylogenetic networks

Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal gene transfer. Indeed, each phylogenetic network embeds a collection of phylogenetic trees. Referring to the collection of trees that a given phylogenetic network $N$ embeds as the display set of $N$, several questions in the context of the display set of $N$ have recently been analyzed. For example, the widely studied Tree-Containment problem asks if a given phylogenetic tree is contained in the display set of a given network. The focus of this paper are two questions that naturally arise in comparing the display sets of two phylogenetic networks. First, we analyze the problem of deciding if the display sets of two phylogenetic networks have a tree in common. Surprisingly, this problem turns out to be NP-complete even for two temporal normal networks. Second, we investigate the question of whether or not the display sets of two phylogenetic networks are equal. While we recently showed that this problem is polynomial-time solvable for a normal and a tree-child network, it is computationally hard in the general case. In establishing hardness, we show that the problem is contained in the second level of the polynomial-time hierarchy. Specifically, it is $Π_2^P$-complete. Along the way, we show that two other problems are also $Π_2^P$-complete, one of which being a generalization of Tree-Containment.

math.CO↗

arXiv2019

Display sets of normal and tree-child networks

Phylogenetic trees canonically arise as embeddings of phylogenetic networks. We recently showed that the problem of deciding if two phylogenetic networks embed the same sets of phylogenetic trees is computationally hard, \blue{in particular, we showed it to be $Π^P_2$-complete}. In this paper, we establish a polynomial-time algorithm for this decision problem if the initial two networks consists of a normal network and a tree-child network. The running time of the algorithm is quadratic in the size of the leaf sets.

math.CO↗

arXiv2019

Attaching leaves and picking cherries to characterise the hybridisation number for a set of phylogenies

Throughout the last decade, we have seen much progress towards characterising and computing the minimum hybridisation number for a set P of rooted phylogenetic trees. Roughly speaking, this minimum quantifies the number of hybridisation events needed to explain a set of phylogenetic trees by simultaneously embedding them into a phylogenetic network. From a mathematical viewpoint, the notion of agreement forests is the underpinning concept for almost all results that are related to calculating the minimum hybridisation number for when |P|=2. However, despite various attempts, characterising this number in terms of agreement forests for |P|>2 remains elusive. In this paper, we characterise the minimum hybridisation number for when P is of arbitrary size and consists of not necessarily binary trees. Building on our previous work on cherry-picking sequences, we first establish a new characterisation to compute the minimum hybridisation number in the space of tree-child networks. Subsequently, we show how this characterisation extends to the space of all rooted phylogenetic networks. Moreover, we establish a particular hardness result that gives new insight into some of the limitations of agreement forests.

q-bio.PE↗

arXiv2018

Excluded minors are almost fragile

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb P$, and let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is bounded relative to $N$, or, up to replacing $M$ by a $Δ$-$Y$-equivalent excluded minor, we can choose a pair of elements $\{a,b\}$ such that either $M\backslash \{a,b\}$ is $N$-fragile, or $M^* \backslash \{a,b\}$ is $N^*$-fragile.

math.CO↗

arXiv2018

Tree-based networks: characterisations, metrics, and support trees

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree $T$, called a base tree, and adding arcs between arcs of $T$. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network $N$, we characterise the support trees of $N$, that is, the tree-based embeddings of $N$. We use this characterisation to give an explicit formula for the number of support trees of $N$ when $N$ is binary. This formula is written in terms of the components of a bipartite graph.

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arXiv2018

Matroids with a cyclic arrangement of circuits and cocircuits

For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $σ$ such that every set of $t-1$ consecutive elements in $σ$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $σ$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element circuits and cocircuits appear in swirls. Furthermore, we show that any appropriate concatenation $Φ$ of $σ$ is a flower. If $t$ is odd, then $Φ$ is a daisy, but if $t$ is even, then, depending on $M$, it is possible for $Φ$ to be either an anemone or a daisy.

math.CO↗

arXiv2017

A universal tree-based network with the minimum number of reticulations

A tree-based network $\mathcal N$ on $X$ is universal if every rooted binary phylogenetic $X$-tree is a base tree for $\mathcal N$. Hayamizu and, independently, Zhang constructively showed that, for all positive integers $n$, there exists an universal tree-based network on $n$ leaves. For all $n$, Hayamizu's construction contains $Θ(n!)$ reticulations, while Zhang's construction contains $Θ(n^2)$ reticulations. A simple counting argument shows that an universal tree-based network has $Ω(n\log n)$ reticulations. With this in mind, Hayamizu as well as Steel posed the problem of determining whether or not such networks exists with $O(n\log n)$ reticulations. In this paper, we show that, for all $n$, there exists an universal tree-based network on $n$ leaves with $O(n\log n)$ reticulations.

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