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 2-ultrametrics. 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 quasi-ultrametric 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=2k=2) and quasi-ultrametric can be performed in polynomial time.