267 research outputs found
On Patchworks and Hierarchies
Motivated by questions in biological classification, we discuss some
elementary combinatorial and computational properties of certain set systems
that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample
patchworks' and 'saturated patchworks' and also outline how these concepts
relate to an apparently new 'duality theory' for cluster systems that is based
on the fundamental concept of 'compatibility' of clusters.Comment: 17 pages, 2 figure
A matroid associated with a phylogenetic tree
A (pseudo-)metric D on a finite set X is said to be a `tree metric' if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y âX, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is 13; up to canonical isomorphism 13; uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree's edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (`lasso') these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the `tight edge-weight lassos' for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T
A Hall-type theorem for triplet set systems based on medians in trees
Given a collection \C of subsets of a finite set , let \bigcup \C =
\cup_{S \in \C}S. Philip Hall's celebrated theorem \cite{hall} concerning
`systems of distinct representatives' tells us that for any collection \C of
subsets of there exists an injective (i.e. one-to-one) function f: \C \to
X with for all S \in \C if and and only if \C satisfies the
property that for all non-empty subsets \C' of \C we have |\bigcup \C'|
\geq |\C'|. Here we show that if the condition
|\bigcup \C'| \geq |\C'| is replaced by the stronger condition |\bigcup
\C'| \geq |\C'|+2, then we obtain a characterization of this condition for a
collection of 3-element subsets of in terms of the existence of an
injective function from \C to the vertices of a tree whose vertex set
includes and that satisfies a certain median condition. We then describe an
extension of this result to collections of arbitrary-cardinality subsets of
.Comment: 6 pages, no figure
Characterizing block graphs in terms of their vertex-induced partitions
Block graphs are a generalization of trees that arise in areas such as metric graph theory, molecular graphs, and phylogenetics. Given a finite connected simple graph with vertex set and edge set , we will show that the (necessarily unique) smallest block graph with vertex set whose edge set contains is uniquely determined by the -indexed family \Pp_G =\big(\pi_v)_{v \in V} of the partitions of the set into the set of connected components of the graph . Moreover, we show that an arbitrary -indexed family \Pp=(\p_v)_{v \in V} of partitions \p_v of the set is of the form \Pp=\Pp_G for some connected simple graph with vertex set as above if and only if, for any two distinct elements , the union of the set in \p_v that contains and the set in \p_u that contains coincides with the set , and \{v\}\in \p_v holds for all . As well as being of inherent interest to the theory of block graphs,these facts are also useful in the analysis of compatible decompositions of finite metric spaces
The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, λ-rings, and the universal ring of Witt vectors
AbstractIt is shown that well-known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the Burnside ring of the infinite cyclic group on the one hand and Grothendieck's ring of formal power series with constant term 1 as well as the universal ring of Witt vectors on the other hand
10231 Abstracts Collection -- Structure Discovery in Biology: Motifs, Networks & Phylogenies
From 06.06. to 11.06.2010, the Dagstuhl Seminar 10231 ``Structure Discovery in Biology: Motifs, Networks & Phylogenies \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Les Pavages d'Anges et de Diables
On utilise la mĂ©thode des symboles de Delaney pour classifier Ă lâaide de Iâordinateur, Ă homĂ©omorphisme Ă©quivariant prĂšs, tous les pavages pĂ©riodiques du plan dont les pavĂ©s peuvent ĂȘtre colories de noir et de blanc de telle maniĂšre que les pavĂ©s se partageant une arĂȘte soient de couleurs diffĂ©rentes, que le groupe de symĂ©trie agisse de faGon transitive sur les pavĂ©s noirs, que tout pavĂ© possĂšde au moins trois arĂȘtes et que de chaque sommet soient issues au moins trois arĂȘtes.The method of Delaney symbols is used to classify by a computer program all periodic tilings of the Euclidean plane up to equivariant homeomorphisms for which the tiles can be coloured by black and white such that tiles sharing an edge have different colours, the symmetry group acts transitively on the black tiles, every tile has at least three edges and from every vertex at least three edges originate.Peer Reviewe
Searching for Realizations of Finite Metric Spaces in Tight Spans
An important problem that commonly arises in areas such as internet
traffic-flow analysis, phylogenetics and electrical circuit design, is to find
a representation of any given metric on a finite set by an edge-weighted
graph, such that the total edge length of the graph is minimum over all such
graphs. Such a graph is called an optimal realization and finding such
realizations is known to be NP-hard. Recently Varone presented a heuristic
greedy algorithm for computing optimal realizations. Here we present an
alternative heuristic that exploits the relationship between realizations of
the metric and its so-called tight span . The tight span is a
canonical polytopal complex that can be associated to , and our approach
explores parts of for realizations in a way that is similar to the
classical simplex algorithm. We also provide computational results illustrating
the performance of our approach for different types of metrics, including
-distances and two-decomposable metrics for which it is provably possible
to find optimal realizations in their tight spans.Comment: 20 pages, 3 figure
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