Sub-dominant theory in numerical taxonomy

Sub-dominant theory provides efficient tools for clustering. However it classically works only for ultrametrics and ad hoc extensions like Jardine and Sibson’s 2ultrametrics. In this paper we study the extension of the notion of sub-dominant to other distance models in classification accounting for overlapping clusters. We prove that a given dissimilarity admits one and only one lower-maximal quasiultrametric and one and only one lower-maximal weak k-ultrametric. In addition, we also prove the existence of (several) lower-maximal strongly Robinsonian dissimilarities. The construction of the lower-maximal weak k-ultrametric (for k = 2) and quasi-ultrametric can be performed in polynomial time.