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Encoding and ordering X-cactuses
Abstract
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 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 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 provide a new approach to solving topical problems in phylogenetic network theory such as deriving consensus networks.
... Figure 5. A spinal network with cover 1, 3 | 5 | 2, 6 | 5, 7 | 4, 6, 8 | 7, 9. Note that n = 4 and the cover has one set in [4], two in [5], three in [6], four in [7], five in [8], and six in [9]. There is a path from the elements of the set that is in [4], namely 1 and 3, to the root, that traverses every non-leaf vertex. ...
... What seems like a fairly straightforward idea in a paper by Diaconis and Holmes (that rooted binary phylogenetic trees correspond to perfect matchings [4]), itself building on an elegant but simple way to label internal vertices [6], was loosened slightly to yield a correspondence between phylogenetic forests and all partitions of finite sets, and a raft of interesting questions in semigroup theory [8]. This subtle twist of an idea, like something from a Philip Pullman novel [18], seems to have opened up further opportunities that, with a further gentle twist, opened a new canvas on which to draw phylogenetic networks [7]. Capturing on this canvas the features that define different network classes provided the underlying motivation for this paper. ...
It was recently shown that a large class of phylogenetic networks, the `labellable' networks, is in bijection with the set of `expanding' covers of finite sets. In this paper, we show how several prominent classes of phylogenetic networks can be characterised purely in terms of properties of their associated covers. These classes include the tree-based, tree-child, orchard, tree-sibling, and normal networks.
... For instance, a classical encoding of trees is the Newick format, which records clusters (descendents of internal vertices) in a structured, in-line notation (used in [7]), and there are encodings using sequences of integers [1,13]. In networks, encodings may require additional structure, such as the use of 'circular' permutations for a class of unrooted phylogenetic networks [8]. ...
... It is expanding, because: it has two subsets of [n] = [5] (the definition of expanding requires at least one); three subsets of [6] (needs at least two); three subsets of [7] (needs three); four subsets of [8]; five subsets of [9]; six subsets of [10]; seven subsets of [11]; and eight subsets of [12]. See Fig. 4 for an illustration of the network constructed from this cover, using Algorithm 2. ...
A bstract
Phylogenetic networks are mathematical representations of evolutionary history that are able to capture both tree-like evolutionary processes (speciations), and non-tree-like “reticulate” processes such as hybridization or horizontal gene transfer. The additional complexity that comes with this capacity, however, makes networks harder to infer from data, and more complicated to work with as mathematical objects.
In this paper we define a new, large class of phylogenetic networks, that we call labellable , and show that they are in bijection with the set of “expanding covers” of finite sets. This correspondence is a generalisation of the encoding of phylogenetic forests by partitions of finite sets. Labellable networks can be characterised by a simple combinatorial condition, and we describe the relationship between this large class and other commonly studied classes. Furthermore, we show that all phylogenetic networks have a quotient network that is labellable.
... For instance, a classical encoding of trees is the Newick format, which records clusters (descendents of internal vertices) in a structured, in-line notation (used in Felsenstein (1989)), and there are encodings using sequences of integers (Bandelt and Dress (1986); James Rohlf (1983)). In networks, classes can be encoded by specific substructures, such as 'trinets' (for tree-child networks) (van Iersel and Moulton 2014), or additional structures such as the use of 'circular' permutations for a class of unrooted phylogenetic networks (Francis et al. 2023). A sub-class of (unlabelled) tree-child networks on n leaves can also be encoded by words of a certain type over an alphabet of size n (Fuchs et al. 2021). ...
Phylogenetic networks are mathematical representations of evolutionary history that are able to capture both tree-like evolutionary processes (speciations) and non-tree-like 'reticulate' processes such as hybridization or horizontal gene transfer. The additional complexity that comes with this capacity, however, makes networks harder to infer from data, and more complicated to work with as mathematical objects. In this paper, we define a new, large class of phylogenetic networks, that we call labellable, and show that they are in bijection with the set of 'expanding covers' of finite sets. This correspondence is a generalisation of the encoding of phylogenetic forests by partitions of finite sets. Labellable networks can be characterised by a simple combinatorial condition, and we describe the relationship between this large class and other commonly studied classes. Furthermore, we show that all phylogenetic networks have a quotient network that is labellable.
It was recently shown that a large class of phylogenetic networks, the ‘labellable’ networks, is in bijection with the set of ‘expanding’ covers of finite sets. In this paper, we show how several prominent classes of phylogenetic networks can be characterised purely in terms of properties of their associated covers. These classes include the tree-based, tree-child, orchard, tree-sibling, and normal networks. In the opposite direction, we give an example of how a restriction on the set of expanding covers can define a new class of networks, which we call ‘spinal’ phylogenetic networks.
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(n3) time, where n is the number of points in the space.
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 model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems.
Phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees, such networks canonically induce collections of phylogenetic trees, clusters, and triplets, respectively. Thus it is not surprising that many network approaches aim to reconstruct a phylogenetic network from such collections. Related to the well-studied perfect phylogeny problem, the following question is of fundamental importance in this context: When does one of the above collections encode (i.e. uniquely describe) the network that induces it? For the large class of level-1 (phylogenetic) networks we characterize those level-1 networks for which an encoding in terms of one (or equivalently all) of the above collections exists. In addition, we show that three known distance measures for comparing phylogenetic networks are in fact metrics on the resulting subclass and give the diameter for two of them. Finally, we investigate the related concept of indistinguishability and also show that many properties enjoyed by level-1 networks are not satisfied by networks of higher level.
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.
Phylogenetic networks are a generalization of evolutionary or phylogenetic trees that are used to represent the evolution of species which have undergone reticulate evolution. In this paper we consider spaces of such networks defined by some novel local operations that we introduce for converting one phylogenetic network into another. These operations are modeled on the well-studied nearest-neighbor interchange operations on phylogenetic trees, and lead to natural generalizations of the tree spaces that have been previously associated to such operations. We present several results on spaces of some relatively simple networks, called level-1 networks, including the size of the neighborhood of a fixed network, and bounds on the diameter of the metric defined by taking the smallest number of operations required to convert one network into another. We expect that our results will be useful in the development of methods for systematically searching for optimal phylogenetic networks using, for example, likelihood and Bayesian approaches.
We investigate the topology and combinatorics of a topological space called the edge-product space that is generated by the set of edge-weighted finite labelled trees. This space arises by multiplying the weights of edges on paths in trees, and is closely connected to tree-indexed Markov processes in molecular evolutionary biology. In particular, by considering combinatorial properties of the Tuffley poset of labelled forests, we show that the edge-product space has a regular cell decomposition with face poset equal to the Tuffley poset.
Given a family ofbinarycharacters defined on a setX, a problem arising in biological and linguistic classification is to decide whether there is a tree structure onXwhich is “compatible” with this family. A fundamental result from hierarchical clustering theory states that there exists a tree structure onXfor such a family if and only if any two of the characters arecompatible. In this paper, we prove a generalization of this result. Namely, we show that given a family ofmulti-statecharacters onXwhich we denote by χ, there exists a tree structure onX, called an (X, χ)-tree, which is “compatible” with χ if and only if any two of the characters arestrongly compatible. To prove this result, we introduce the concept ofblock systems, set theoretical structures which arise naturally from, amongst other things,block graphs, and the related concepts ofblock interval systemsand Δ-systems.
We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn×R. This leads to a proof that the link of the origin in Tn is homeomorphic to the order complex of the proper part of the partition lattice Πn.
We show that for allk⩾1 andn⩾0 the simplicial complexes (k)nof all leaf-labelled trees withnk+2 leaves and all interior vertices of degreeskl+2 (l⩾1) are shellable. This yields a direct combinatorial proof that they are Cohen–Macaulay and that their homotopy types are wedges of spheres.
A topological approach to tree (re-)construction
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