Quasi-ultrametrics and their 2-ball hypergraphs Original Research Article

A quasi-ultrametric is a dissimilarity function satisfying two conditions involving 2-balls (intersections of pairs of balls): the inclusion and the diameter conditions. The 2-ball hypergraph of any quasi-ultrametric is shown to be triangle-free. Adding some classical constraints to such a hypergraph, we obtain the so-called quasi-hierarchies. Moreover, the well-known bijection between indexed hierarchies and ultrametrics (Benzécri, 1973; Jardine et al., 1967 and Johnson, 1967) is extended to indexed quasi-hierarchies and quasi-ultrametrics. This result unifies the previously-mentioned bijection and the one relating indexed pseudo-hierarchies to strongly Robinsonian dissimilarities (Fichet, 1986). It is also generalized in terms of stratified quasi-hierarchies and quasi-ultrametric preordonnances.