c Birkh¨auser Verlag, Basel, 2001
Annals of Combinatorics 5 (2001) 99-112
Annals of Combinatorics
0218-0006/01/010099-14$1.50+0.20/0
Totally Split-Decomposable Metrics of Combinatorial Dimension Two A. Dress1∗ , K.T. Huber2† , and V. Moulton3‡ 1FSPM-Strukturbildungsprozesse, University of Bielefeld, D-33501 Bielefeld, Germany
[email protected] 2Institute of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North,
New Zealand 3FMI, Mid Sweden University, Sundsvall, S 851-70, Sweden
Received November 30, 1999 AMS Subject Classification: 04A03, 04A20, 05C99, 52B99, 92B99 Abstract. The combinatorial dimension of a metric space (X, d), denoted by dimcombin (d), arises naturally in the subject of T-theory, and, in case X is finite, corresponds with the (topological) dimension of the tight span associated to d. Metric spaces of combinatorial dimension at most one are well understood; they are precisely the metric spaces that can be embedded into R-trees. However, the structure of metric spaces of higher dimension is not so well understood. Indeed, even 2-dimensional finite metric spaces can have a rich structure. In this paper, we study finite metric spaces of combinatorial dimension two that are, in addition, totally split decomposable. In particular, we give several characterizations of such metrics derived through the study of a certain map that relates the tight span of a totally splitdecomposable metric with its Buneman complex. Keywords: metric space, consistent metric, tight span, combinatorial dimension, Buneman complex, split system, compatibility, incompatibility, weakly compatible, 2-compatible, k-compatible, 3-crossfree
References 1. H.-J. Bandelt and A. Dress, A canonical decomposition theory for metrics on a finite set, Adv. Math. 92 (1992) 47–105. 2. A. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math. 53 (1984) 321–402. While the results presented here were achieved the author worked as a Distinguished Visiting Professor at the Dept. of Chem. Eng., City College, CUNY, New York. † The author thanks the New Zealand Marsden Fund for its financial support, and the Forschungsschwerpunkt Strukturbildungsprozesse for hosting her during part of this work. ‡ The author thanks the Swedish National Research Council (NFR) for its support (grant# M12342-300).
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3. A. Dress, K. Huber, and V. Moulton, Some variations on a theme by Buneman, Ann. Combin. 1 (1997) 339–352. 4. A. Dress, K.T. Huber, and V. Moulton, A comparison between two distinct continuous models in projective cluster theory: the median and the tight-span construction, Ann. Combin. 2 (1998) 299–311. 5. A. Dress, K.T. Huber, and V. Moulton, An exceptional split geometry, Ann. Combin. 4 (2000) 1–11. 6. A. Dress, K.T. Huber, and V. Moulton, An explicit computation of the injective hull of certain finite metric spaces in terms of their associated Buneman complex, manuscript, Bielefeld, 1999. 7. A. Dress, K.T. Huber, and V. Moulton, Affine maps that induce polytope isomorphisms, Discrete Comput. Geom. 24 (2000) 49-60. 8. A. Dress, K.T. Huber, J.H. Koolen, and V. Moulton, Six points suffice: how to check for metric consistency, Europ. J. Combin., in press. 9. A. Dress, J. Koolen, and V. Moulton, 4n − 10, manuscript, Bielefeld, 1999. 10. A. Karzanov, Combinatorial methods to solve cut-determined multiflow problems, In: Combinatorial Methods for Flow Problems, No. 3, A. Karzanov Ed., Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, 1979, pp. 6–69 (in Russian). 11. A. Karzanov, Metrics with finite sets of primitive extensions, Ann. Combin. 2 (1998) 213– 243. 12. M. Moeller, Mengen schwach vertr¨aglicher splits, Ph.D. Thesis, Bielefeld, 1990.
