A classification of the six-point prime metrics

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.