90 research outputs found
Polyhedral geometry of Phylogenetic Rogue Taxa
It is well known among phylogeneticists that adding an extra taxon (e.g.
species) to a data set can alter the structure of the optimal phylogenetic tree
in surprising ways. However, little is known about this "rogue taxon" effect.
In this paper we characterize the behavior of balanced minimum evolution (BME)
phylogenetics on data sets of this type using tools from polyhedral geometry.
First we show that for any distance matrix there exist distances to a "rogue
taxon" such that the BME-optimal tree for the data set with the new taxon does
not contain any nontrivial splits (bipartitions) of the optimal tree for the
original data. Second, we prove a theorem which restricts the topology of
BME-optimal trees for data sets of this type, thus showing that a rogue taxon
cannot have an arbitrary effect on the optimal tree. Third, we construct
polyhedral cones computationally which give complete answers for BME rogue
taxon behavior when our original data fits a tree on four, five, and six taxa.
We use these cones to derive sufficient conditions for rogue taxon behavior for
four taxa, and to understand the frequency of the rogue taxon effect via
simulation.Comment: In this version, we add quartet distances and fix Table 4
Tropical Convexity
The notions of convexity and convex polytopes are introduced in the setting
of tropical geometry. Combinatorial types of tropical polytopes are shown to be
in bijection with regular triangulations of products of two simplices.
Applications to phylogenetic trees are discussed.
Theorem 29 and Corollary 30 in the paper, relating tropical polytopes to
injective hulls, are incorrect. See the erratum at
http://www.math.uiuc.edu/documenta/vol-09/vol-09-eng.html .Comment: 20 pages, 6 figure
Symmetry in Graph Theory
This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view
A combinatorial non-positive curvature I: weak systolicity
We introduce the notion of weakly systolic complexes and groups, and initiate
regular studies of them. Those are simplicial complexes with
nonpositive-curvature-like properties and groups acting on them geometrically.
We characterize weakly systolic complexes as simply connected simplicial
complexes satisfying some local combinatorial conditions. We provide several
classes of examples --- in particular systolic groups and CAT(-1) cubical
groups are weakly systolic. We present applications of the theory, concerning
Gromov hyperbolic groups, Coxeter groups and systolic groups.Comment: 35 pages, 1 figur
Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective
Phylogenetic trees are the fundamental mathematical representation of
evolutionary processes in biology. As data objects, they are characterized by
the challenges associated with "big data," as well as the complication that
their discrete geometric structure results in a non-Euclidean phylogenetic tree
space, which poses computational and statistical limitations. We propose and
study a novel framework to study sets of phylogenetic trees based on tropical
geometry. In particular, we focus on characterizing our framework for
statistical analyses of evolutionary biological processes represented by
phylogenetic trees. Our setting exhibits analytic, geometric, and topological
properties that are desirable for theoretical studies in probability and
statistics, as well as increased computational efficiency over the current
state-of-the-art. We demonstrate our approach on seasonal influenza data.Comment: 28 pages, 5 figures, 1 tabl
Polyhedral computational geometry for averaging metric phylogenetic trees
This paper investigates the computational geometry relevant to calculations
of the Frechet mean and variance for probability distributions on the
phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature developed by Sturm. We
show that the combinatorics of geodesics with a specified fixed endpoint in
tree space are determined by the location of the varying endpoint in a certain
polyhedral subdivision of tree space. The variance function associated to a
finite subset of tree space has a fixed algebraic formula within
each cell of the corresponding subdivision, and is continuously differentiable
in the interior of each orthant of tree space. We use this subdivision to
establish two iterative methods for producing sequences that converge to the
Frechet mean: one based on Sturm's Law of Large Numbers, and another based on
descent algorithms for finding optima of smooth functions on convex polyhedra.
We present properties and biological applications of Frechet means and extend
our main results to more general globally nonpositively curved spaces composed
of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5,
added counter example for polyhedrality of vistal subdivision in general
CAT(0) cubical complexes; v1: 43 pages, 5 figure
On the edge metric dimension of some classes of cacti
The cactus graph has many practical applications, particularly in radio communication systems. Let be a finite, undirected, and simple connected graph, then the edge metric dimension of is the minimum cardinality of the edge metric generator for (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices of a connected graph , for any edge , we referred to the -vector (ordered -tuple), as the edge metric representation of with respect to . In this regard, is an edge metric generator for if, and only if, for every pair of distinct edges implies . In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: and in terms of the number of cycles and the number of paths
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