Embedding Topological Median Algebras in Products of Dendrons

Bandelt, Hans‐Jürgen; Vel, M. van de

doi:10.1112/plms/s3-58.3.439

Cited by 34 publications

(58 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, every partition of the character set into sets of pairwise compatible characters is a shelling centered at the zero vector and vice versa. The shellings centered at the 14 zero vector are therefore in one-to-one correspondence with the colorings of the graph G (Bandelt and van de Vel, 1989). For any shelling with another center, the deepest shell consists of a single character, which could as well become the outer shell instead, so that this shifted shelling would now be centered at the zero vector.…”

Section: Shellingmentioning

confidence: 97%

Median Networks: Speedy Construction and Greedy Reduction, One Simulation, and Two Case Studies from Human mtDNA

Bandelt

Macaulay

Richards

2000

Molecular Phylogenetics and Evolution

184

136

View full text Add to dashboard Cite

Molecular data sets characterized by few phylogenetically informative characters with a broad spectrum of mutation rates, such as intraspecific controlregion sequence variation of human mitochondrial DNA (mtDNA), can be usefully visualized in the form of median networks. Here we provide a step-by-step guide to the construction of such networks by hand. We improve upon a previously implemented algorithm by outlining an efficient parametrized strategy amenable to large data sets, greedy reduction, which makes it possible to reconstruct some of the confounding recurrent mutations. This entails some postprocessing as well, which assists in capturing more parsimonious solutions. To simplify the creation of the resulting network by hand, we describe a speedy approach to network construction, based on a careful planning of the processing order. A coalescent simulation tailored to human mtDNA variation in Eurasia testifies to the usefulness of reduced median networks, while highlighting notorious problems faced by all phylogenetic methods in this context. Finally, we discuss two case studies involving the comparison of characters in the two hypervariable segments of the human mtDNA control region in the light of the worldwide control-region sequence database, as well as additional restriction fragment length polymorphism information. We conclude that only a minority of the mutations that hit the second segment occur at sites that would have a mutation rate comparable to those at most sites in the first segment. Discarding the known "noisy" sites of the second segment enhances the analysis.

show abstract

Section: Shellingmentioning

confidence: 97%

Median Networks: Speedy Construction and Greedy Reduction, One Simulation, and Two Case Studies from Human mtDNA

Bandelt

Macaulay

Richards

2000

Molecular Phylogenetics and Evolution

184

136

View full text Add to dashboard Cite

show abstract

“…Theorem 6.1 essentially facilitates this. Moreover, most of such properties have been already known for median graphs; [11,14,15,27,42,43,54,55] are just a few sources. On the other hand, the CAT(0) property sheds a new light on median complexes.…”

Section: Properties Of Median Complexesmentioning

confidence: 98%

“…Median graphs and the related median structures have many nice characterizations and properties, investigated by several authors; [5,7,11,14,15,35,42,43,54,55,56] is a sample of papers on this subject.…”

Section: A Graph G Is Modular If and Only If It Is Triangle-free Anmentioning

confidence: 99%

Graphs of Some CAT(0) Complexes

Chepoi

2000

Advances in Applied Mathematics

214

251

View full text Add to dashboard Cite

In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov's CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G K is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT (0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1-or 2-fellow traveler property.

show abstract

“…Clearly the simplex graph of the complete graph K n is the hypercube Q n (using the fact that hypercubes are characterized as subset graphs via positions of 1's in the binary representation of Q n ). It is also not surprising that the simplex graph of any graph is always a median graph, as shown in [5]. There it was also proved that a hypercube in κ(G) of maximum dimension d corresponds to some largest clique of order d in G. In fact any maximal hypercube in κ(G) corresponds precisely to some maximal complete subgraph (clique) of G. Furthermore, we find that two maximal hypercubes share an edge in κ(G) whenever the corresponding cliques in G share a vertex (this fact was discovered by Chepoi [17]; for a related study we refer to a commendable survey [3]).…”

Section: Edge-intersection Of Maximal Hypercubesmentioning

confidence: 95%

Cube intersection concepts in median graphs

Brešar

Šumenjak

2009

Discrete Mathematics

View full text Add to dashboard Cite

a b s t r a c tIn this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k ≥ 0, the intersection graph Q k (G)is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices H x and H y in Q k (G) are adjacent whenever the intersection H x ∩ H y contains a subgraph isomorphic to Q k . Characterizations of clique-graphs in terms of these intersection concepts when k > 0, are presented. Furthermore, we introduce the socalled maximal 2-intersection graph of maximal hypercubes of a median graph G, denoted Q m2 (G), whose vertices are maximal hypercubes of G, and two vertices are adjacent if the intersection of the corresponding hypercubes is not a proper subcube of some intersection of two maximal hypercubes. We show that a graph H is diamond-free if and only if there exists a median graph G such that H is isomorphic to Q m2 (G). We also study convergence of median graphs to the one-vertex graph with respect to all these operations.

show abstract

Embedding Topological Median Algebras in Products of Dendrons

Cited by 34 publications

References 14 publications

Median Networks: Speedy Construction and Greedy Reduction, One Simulation, and Two Case Studies from Human mtDNA

Median Networks: Speedy Construction and Greedy Reduction, One Simulation, and Two Case Studies from Human mtDNA

Graphs of Some CAT(0) Complexes

Cube intersection concepts in median graphs

Contact Info

Product

Resources

About