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1. Dress, A.

et al.

Huber, K. T.

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Moulton, Vincent

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Antipodal metrics and split systems2002In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 23, no 2, p. 187-200Article in journal (Refereed)

Abstract [en]

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.

In many areas of data analysis, it is desirable to have tools at hand for analyzing the structure of distance tables-or, in more mathematical terms, of finite metric spaces. One such tool, known as split decomposition theory has proven particularly useful in this respect. Tbe class of so-called totally decomposable metrics forms a cornerstone for this theory, and much work has been devoted to their study. Recently, it has become apparent that a particular subclass of these metrics, the consistent metrics, are also of fundamental importance. In this paper, we give a six-point characterization of consistent metrics amongst the totally decomposable ones.

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

On line arrangements in the hyperbolic plane2002In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 23, no 5, p. 549-557Article in journal (Refereed)

Abstract [en]

Given a finite collection L of lines in the hyperbolic plane H, we denote by k = k(L) its Karzanov number, i.e., the maximal number of pairwise intersecting lines in L, and by C(L) and n = n(L) the set and the number, respectively, of those points at infinity that are incident with at least one line from L. By using purely combinatorial properties of cyclic seta:, it is shown that #L less than or equal to 2nk - ((2k+1)(2)) always holds and that #L equals 2nk - ((2k+1)(2)) if and only if there is no collection L' of lines in H with L subset of or equal to L', k(L') = k(L) and C(L') = C(L).

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Hyperbolic bridged graphs2002In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 23, no 6, p. 683-699Article in journal (Refereed)

Abstract [en]

Given a connected graph G, we take, as usual, the distance xy between any two vertices x, y of G to be the length of some geodesic between x and y. The graph G is said to be delta-hyperbolic, for some 3 : 0, if for all vertices x, y, u, v in G the inequality xy + uv :5 max{xu + yv, xv + yu} + delta holds, and G is bridged if it contains no finite isometric cycles of length four or more. In this paper, we will show that a finite connected bridged graph is 1-hyperbolic if and only if it does not contain any of a list of six graphs as an isometric subgraph.

The notion of a coherent decomposition of a metric on a finite set has proven fruitful, with applications to areas such as the geometry of metric cones and bioinformatics. In order to obtain a deeper insight into these decompositions it is important to improve our knowledge of those metrics which cannot be coherently decomposed in a non-trivial way, i.e.,the prime metrics. In this paper we classify the prime metrics on six points. (C) 2000 Academic Press.