ISSN:
0219-3094
Keywords:
Keywords: split system, incompatible, split system, incompatibility, weakly compatible split system, weak compatibility, T-theory, tight span, Buneman complex, metrics, finite metric spaces
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. In view of results obtained in split decomposition theory, it is of some interest to investigate the structure of weakly compatible split systems. A particular class of such split systems — the so-called octahedral split systems — can be constructed as follows: Given a set X together with a surjective map $ \phi:X\twoheadrightarrow V $ onto the six-element set V of vertices of an octahedron, form the four bipartitions $ X = A_i \dot{\cup} B_i $ (i = 1, 2, 3, 4) of X obtained by first partitioning V in all four possible ways into two disjoint 3-subsets U i and W i (i = 1, 2, 3, 4) so that the vertices in both U i and W i form an equilateral triangle, and then taking their pre-images A i : = $ \phi $ -1(U i ) and B i : = $ \phi $ -1(W i ) (i = 1, 2, 3, 4).¶In this note, it will be shown that a weakly compatible split system $ {\cal S} $ is octahedral if and only if it is not circular while, simultaneously, any two splits in $ {\cal S} $ are incompatible. This result appeared originally in Martina Moeller's Ph.D. thesis. Here, we give an alternative proof based on the close relationship between weakly compatible split systems and weak hierarchies.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00001271
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