arXiv subjects

Yukihiro Murakami

Publications and source records attributed to Yukihiro Murakami.

At least 19 recordsLinked to original sources

arXiv2026

How many cherry-picking sequences are needed to reduce all subtrees of a phylogenetic tree?

Phylogenetic networks are graphs that represent the evolutionary history of species. Recently, the class of orchard phylogenetic networks, which can be reduced by so-called cherry-picking sequences, has gained attention for its computational and biological aspects. In this paper, we study a fundamental question on orchards and their cherry-picking sequences by considering the CoveringNumber problem: given an orchard network $N$, how many cherry-picking sequences are needed to reduce all subnetworks of $N$? We initiate this study by considering the problem for trees. We then show that the covering number can be computed for binary trees recursively using a similar but more fine-grained notion of survival covering number. We also give a recursive formula for the survival covering number of non-binary trees. However, computing the covering number for non-binary trees appears to be considerably more challenging. For this case, we show that the covering number of star trees (whose root is adjacent to all leaves) is equivalent to the so-called SubsetConnectivity problem, which we introduce in this paper. Finally, we show that if there is no restriction on the sequence length, a single sequence of minimum length $\binom{n}{2}$ suffices to reduce all subtrees of a tree on $n$ leaves.

math.CO↗

arXiv2026

Metropolis-Hastings Sampling of Phylogenetic Networks: Correcting for Symmetries

In phylogenetics, Metropolis-Hastings methods are commonly used to sample phylogenetic trees or networks, for example from Bayesian posteriors. These methods generally use transitions that distinguish all nodes involved, and thus require fully labelled representations of phylogenetic networks. We argue that sampling leaf-labelled phylogenetic networks demands a correction for the number of fully labelled representatives of a leaf-labelled network, or, equivalently, for its internal symmetry. Without correction, there is a danger of undersampling networks with internal symmetries. We show that this correction can be realized by a quotient construction on the Metropolis-Hastings Markov chain, which, in practice, requires the calculation of the size of the network's automorphism group. Using $μ$-vectors, we show that the automorphism group is trivial for orchard networks, and thus also for tree-child networks and trees. This implies that a correction for symmetry is not needed when sampling only from such network classes. More generally, using our Python implementation of the algorithms in this paper, we show that using $μ$-vectors can significantly speed up calculations of automorphism group sizes and thus of Metropolis-Hastings sampling of leaf-labelled networks.

cs.DM↗

arXiv2026

Proximity Measures for Classes of Phylogenetic Networks

Phylogenetic networks are used to represent the evolutionary history of species. Due to biological interpretations and computational advantages, researchers have focused on restricted classes of phylogenetic networks, such as tree-child, orchard, and tree-based. These classes capture different notions of tree-likeness: tree-child networks require every internal vertex to have a taxon reachable by a tree path, orchard networks are trees with horizontal arcs (for modelling histories rife with horizontal gene transfers), and tree-based networks are trees with additional (not-necessarily horizontal) arcs. A natural question to ask is ``how far is a given network from belonging to a particular class?'' This motivates the study of proximity measures, which measure the minimum number of graph modifications required to transform a network into one belonging to a particular class. In this paper, we consider three proximity measures based on leaf addition, valid arc deletion, and arc deletion. We study pairwise comparability of the proximity measures, prove complexity results, and derive extremal bounds for the classes of tree, tree-child, orchard, and tree-based networks.

math.CO↗

arXiv2026

List $3$-coloring $C_4$-free graphs of diameter-$2$ in polynomial-time

We show that list $3$-coloring a~$C_4$-free graph of diameter-$2$ can be done in polynomial-time. Our algorithm is based on a structural characterization showing that many such graphs are not~$3$-colorable. In particular, we show that~$C_4$-free graphs of diameter-$2$ without universal vertices, where the maximum degree is at least~$17$, are not~$3$-colorable.

math.CO↗

arXiv2026

Burning Graph Powers and Branching Trees

Graph burning is a discrete-time process that models the spread of social contagion. Initially, all vertices are unburned. In each round, one unburned vertex is selected and burned, while any unburned vertex that has a burned neighbour from the previous round also becomes burned. The burning number of a graph is the minimum number of rounds needed to burn the entire graph. In this paper, we study the burning number of graph powers. First, we show that for a connected graph~$G$, its graph power~$G^k$ contains a~$(k+1)^+$-branching tree as a spanning tree. A~$(k+1)^+$-branching tree is one in which all internal vertices have degree at least~$k+1$. We then show that $(k+1)^+$-branching trees on~$n$ vertices have burning number at most $\left\lceil{\sqrt{\frac{4(k-1)n}{k^2}}}~\right\rceil$. As the burning number of a graph is at most the burning number of any of its spanning trees, this gives an upper bound on the burning number of graph powers. We also derive an alternative upper bound on the burning number of~$k^+$-branching trees using the strongest currently known general burning number bound [Bastide et al.]. We then identify the ranges of~$k$ and~$n$ for which our bound outperforms or matches this alternative bound. Finally, we show that~$b(G^k) \le (1+o(1))\sqrt{n/k}$ based on the asymptotic burning number bound of Norin and Turcotte.

math.CO↗

arXiv2026

Note on the Maximum Number of Trees Displayed by a Tree-Child Network

In this note, we show that, for all $n\ge 2$, the number of distinct rooted binary phylogenetic $X$-trees displayed by a binary tree-child network $\mathcal{N}$ on $X$ with $n$ leaves is at most $2^{n-1}-1$ and that this upper bound is sharp. Furthermore, if $\mathcal{N}$ displays exactly $2^{n-1}-1$ such trees, then exactly one rooted binary phylogenetic $X$-tree is displayed twice, and this tree can be canonically found by iteratively replacing a reticulated cherry with a cherry.

math.CO↗

arXiv2025

Metrics for classes of semi-binary phylogenetic networks using $μ$-representations

Phylogenetic networks are useful in representing the evolutionary history of taxa. In certain scenarios, one requires a way to compare different networks. In practice, this can be rather difficult, except within specific classes of networks. In this paper, we derive metrics for the class of \emph{orchard networks} and the class of \emph{strongly reticulation-visible} networks, from variants of so-called \emph{$μ$-representations}, which are vector representations of networks. For both network classes, we impose degree constraints on the vertices, by considering \emph{semi-binary} networks.

q-bio.PE↗

arXiv2024

A Wild Sheep Chase Through an Orchard

Orchards are a biologically relevant class of phylogenetic networks as they can describe treelike evolutionary histories augmented with horizontal transfer events. Moreover, the class has attractive mathematical characterizations that can be exploited algorithmically. On the other hand, undirected orchard networks have hardly been studied yet. Here, we prove that deciding whether an undirected, binary phylogenetic network is an orchard -- or equivalently, whether it has an orientation that makes it a rooted orchard -- is NP-hard. For this, we introduce a new characterization of undirected orchards which could be useful for proving positive results.

q-bio.PE↗

arXiv2024

Fence decompositions and cherry covers in non-binary phylogenetic networks

Reticulate evolution can be modelled using phylogenetic networks. Tree-based networks, which are one of the more general classes of phylogenetic networks, have recently gained eminence for its ability to represent evolutionary histories with an underlying tree structure. To better understand tree-based networks, numerous characterizations have been proposed, based on tree embeddings, matchings, and arc partitions. Here, we build a bridge between two arc partition characterizations, namely maximal fence decompositions and cherry covers. Results on cherry covers have been found for general phylogenetic networks. We first show that the number of cherry covers is the same as the number of support trees (underlying tree structure of tree-based networks) for a given semibinary network. Maximal fence decompositions have only been defined thus far for binary networks (constraints on vertex degrees). We remedy this by generalizing fence decompositions to non-binary networks, and using this, we characterize semi-binary tree-based networks in terms of forbidden structures. Furthermore, we give an explicit enumeration of cherry covers of semi-binary networks, by studying its fence decomposition. Finally, we prove that it is possible to characterize semi-binary tree-child networks, a subclass of tree-based networks, in terms of the number of their cherry covers.

q-bio.PE↗

arXiv2023

The Graph Burning Conjecture is true for trees without degree-2 vertices

Graph burning is a discrete time process which can be used to model the spread of social contagion. One is initially given a graph of unburned vertices. At each round (time step), one vertex is burned; unburned vertices with at least one burned neighbour from the previous round also becomes burned. The burning number of a graph is the fewest number of rounds required to burn the graph. It has been conjectured that for a graph on $n$ vertices, the burning number is at most $\lceil\sqrt{n}\rceil$. We show that the graph burning conjecture is true for trees without degree-2 vertices.

math.CO↗

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

Orienting undirected phylogenetic networks to tree-child network

Phylogenetic networks are used to represent the evolutionary history of species. They are versatile when compared to traditional phylogenetic trees, as they capture more complex evolutionary events such as hybridization and horizontal gene transfer. Distance-based methods such as the Neighbor-Net algorithm are widely used to compute phylogenetic networks from data. However, the output is necessarily an undirected graph, posing a great challenge to deduce the direction of genetic flow in order to infer the true evolutionary history. Recently, Huber et al. investigated two different computational problems relevant to orienting undirected phylogenetic networks into directed ones. In this paper, we consider the problem of orienting an undirected binary network into a tree-child network. We give some necessary conditions for determining the tree-child orientability, such as a tight upper bound on the size of tree-child orientable graphs, as well as many interesting examples. In addition, we introduce new families of undirected phylogenetic networks, the jellyfish graphs and ladder graphs, that are orientable but not tree-child orientable. We also prove that any ladder graph can be made tree-child orientable by adding extra leaves, and describe a simple algorithm for orienting a ladder graph to a tree-child network with the minimum number of extra leaves. We pose many open problems as well.

math.CO↗

arXiv2023

Making a Network Orchard by Adding Leaves

Phylogenetic networks are used to represent the evolutionary history of species. Recently, the new class of orchard networks was introduced, which were later shown to be interpretable as trees with additional horizontal arcs. This makes the network class ideal for capturing evolutionary histories that involve horizontal gene transfers. Here, we study the minimum number of additional leaves needed to make a network orchard. We demonstrate that computing this proximity measure for a given network is NP-hard and describe a tight upper bound. We also give an equivalent measure based on vertex labellings to construct a mixed integer linear programming formulation. Our experimental results, which include both real-world and synthetic data, illustrate the effectiveness of our implementation.

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↗

arXiv2021

Orchard Networks are Trees with Additional Horizontal Arcs

Phylogenetic networks are used in biology to represent evolutionary histories. The class of orchard phylogenetic networks was recently introduced for their computational benefits, without any biological justification. Here, we show that orchard networks can be interpreted as trees with additional \emph{horizontal} arcs. Therefore, they are closely related to tree-based networks, where the difference is that in tree-based networks the additional arcs do not need to be horizontal. Then, we use this new characterization to show that the space of orchard networks is connected under the rNNI rearrangement move, with a diameter of at most $4kn+n\lceil \log_2(n) \rceil +2k+6n-8$.

math.CO↗

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

Distinguishing level-1 phylogenetic networks on the basis of data generated by Markov processes

Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees. These networks are able to incorporate reticulate evolutionary events such as hybridization, introgression, and lateral gene transfer. Recently, network-based Markov models of DNA sequence evolution have been introduced along with model-based methods for reconstructing phylogenetic networks. For these methods to be consistent, the network parameter needs to be identifiable from data generated under the model. Here, we show that the semi-directed network parameter of a triangle-free, level-1 network model with any fixed number of reticulation vertices is generically identifiable under the Jukes-Cantor, Kimura 2-parameter, or Kimura 3-parameter constraints.

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.

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