From hypertrees to arboreal quasi-ultrametrics

Some classical models of clustering (hierarchies, pyramids, etc.) are related to interval hypergraphs. In this paper we study clustering models related to hypertrees which are an extension of interval hypergraphs. We first prove that a hypertree can be characterized by an order on its vertices, this order allowing to find one of its underlying vertex trees. We then focus on clustering models associated to dissimilarity models and prove that if one of the cluster hypergraph, ball hypergraph, or 2-ball hypergraph related to a given dissimilarity is a hypertree, then the two others are also hypertrees. Moreover, we prove that a given dissimilarity admits at least one lower-maximal dissimilarity whose cluster hypergraph is a hypertree, and one and only one lower-maximal quasi-ultrametric whose cluster hypergraph is a hypertree. The construction of the lower-maximal quasi-ultrametric whose cluster hypergraph is a hypertree can be performed in polynomial time.