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 ∈X, 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 $X$, 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 $X$ there exists an injective (i.e. one-to-one) function f: \C \to X with $f(S) \in S$ 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 $X$ in terms of the existence of an injective function from \C to the vertices of a tree whose vertex set includes $X$ and that satisfies a certain median condition. We then describe an extension of this result to collections of arbitrary-cardinality subsets of $X$.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 $G=(V,E)$ with vertex set $V$ and edge set $E\subseteq \binom{V}{2}$, we will show that the (necessarily unique) smallest block graph with vertex set $V$ whose edge set contains $E$ is uniquely determined by the $V$-indexed family \Pp_G =\big(\pi_v)_{v \in V} of the partitions $\pi_v$ of the set $V$ into the set of connected components of the graph $(V,\{e\in E: v\notin e\})$. Moreover, we show that an arbitrary $V$-indexed family \Pp=(\p_v)_{v \in V} of partitions \p_v of the set $V$ is of the form \Pp=\Pp_G for some connected simple graph $G=(V,E)$ with vertex set $V$ as above if and only if, for any two distinct elements $u,v\in V$, the union of the set in \p_v that contains $u$ and the set in \p_u that contains $v$ coincides with the set $V$, and \{v\}\in \p_v holds for all $v \in V$. 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 méthode des symboles de Delaney pour classifier à l’aide de I’ordinateur, à homéomorphisme équivariant près, tous les pavages périodiques du plan dont les pavés peuvent être colories de noir et de blanc de telle manière que les pavés se partageant une arête soient de couleurs différentes, que le groupe de symétrie agisse de faGon transitive sur les pavés noirs, que tout pavé possède au moins trois arêtes et que de chaque sommet soient issues au moins trois arêtes.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 $D$ 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 $D$ and its so-called tight span $T_D$. The tight span $T_D$ is a canonical polytopal complex that can be associated to $D$, and our approach explores parts of $T_D$ 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 $l_1$-distances and two-decomposable metrics for which it is provably possible to find optimal realizations in their tight spans.Comment: 20 pages, 3 figure