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- Dress, A., et al.
(författare)
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Affine maps that induce polyhedral complex isomorphisms
- 2000
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Ingår i: Discrete & Computational Geometry. - 0179-5376 .- 1432-0444. ; 24:1, s. 49-60
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we show that an affine bijection f: T-1 --> T-2 between two polyhedral complexes T-1, T-2, both of which consist of a union of faces of two convex polyhedra P-1 and P-2, necessarily respects the cell-complex structure of T-1 and T-2 inherited from P-1 and P-2, respectively, provided f extends to an affine map from P-1 into P-2. In addition, we present an application of this result within the area of T-theory to obtain a far-reaching generalization of previous results regarding the equivalence of two distinct constructions of the phylogenetic tree associated to "perfect" (that is, treelike) distance data.
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| 2. |
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| 3. |
- Dress, A, et al.
(författare)
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Antipodal metrics and split systems
- 2002
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Ingår i: European Journal of Combinatorics. - : Elsevier BV. - 0195-6698 .- 1095-9971. ; 23:2, s. 187-200
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Tidskriftsartikel (refereegranskat)abstract
- Recall that a metric d on a finite set X is called antipodal if there exists a map sigma : X --> X: x --> (x) over bar so that d(x, (x) over bar) = d(x, y) + d(y, (x) over bar) holds for all x, y epsilon X. Antipodal metrics canonically arise as metrics induced on specific weighted graphs, although their abundance becomes clearer in light of the fact that any finite metric space can be isometrically embedded in a more or less canonical way into an antipodal metric space called its full antipodal extension. In this paper, we examine in some detail antipodal metrics that are, in addition, totally split decomposable. In particular, we give an explicit characterization of such metrics, and prove that-somewhat surprisingly-the full antipodal extension of a proper metric d on a finite set X is totally split decomposable if and only if d is linear or #X = 3 holds.
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| 4. |
- Huber, K T, et al.
(författare)
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An algorithm for constructing local regions in a phylogenetic network
- 2001
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Ingår i: Molecular Phylogenetics and Evolution. - : Elsevier BV. - 1055-7903 .- 1095-9513. ; 19:1, s. 1-8
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Forskningsöversikt (refereegranskat)abstract
- The groupings of taxa in a phylogenetic tree cannot represent all the conflicting signals that usually occur among site patterns in aligned homologous genetic sequences. Hence a tree-building program must compromise by reporting a subset of the patterns, using some discriminatory criterion. Thus, in the worst case, out of possibly a large number of equally good trees, only an arbitrarily chosen tree might be reported by the tree-building program as “The Tree.” This tree might then be used as a basis for phylogenetic conclusions. One strategy to represent conflicting patterns in the data is to construct a network. The Buneman graph is a theoretically very attractive example of such a network. In particular, a characterization for when this network will be a tree is known. Also the Buneman graph contains each of the most parsimonious trees indicated by the data. In this paper we describe a new method for constructing the Buneman graph that can be used for a generalization of Hadamard conjugation to networks. This new method differs from previous methods by allowing us to focus on local regions of the graph without having to first construct the full graph. The construction is illustrated by an example.
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| 5. |
- Huber, K. T., et al.
(författare)
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The relation graph
- 2002
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Ingår i: Discrete Mathematics. - 0012-365X .- 1872-681X. ; 244:1-3, s. 153-166
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Tidskriftsartikel (refereegranskat)abstract
- Given a set R of distinct, non-trivial partitions of a finite set, we define the relation graph G(R) of R. In case R consists only of bipartitions, G(R) is the well-known Buneman graph, a median graph that has applications in the area of phylogenetic analysis., Here we consider properties of the relation graph for general sets of partitions and, in particular, we see that it mimics the behaviour of the Buneman graph by proving the following two theorems:(i) The graph G(R) is a Hamming graph if and only if R is strongly incompatible.(ii) The graph G(R) is a block graph with #R blocks if and only if R is strongly compatible.
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