An order theoretic framework for overlapping clustering
Discrete Mathematics - Special issue: trends in discrete mathematics
Quasi-ultrametrics and their 2-ball hypergraphs
Proceedings of the conference on Discrete metric spaces
Set systems and dissimilarities
European Journal of Combinatorics
Pyramids and weak hierarchies in the ordinal model for clustering
Discrete Applied Mathematics
Pyramids and weak hierarchies in the ordinal model for clustering
Discrete Applied Mathematics
Sub-dominant theory in numerical taxonomy
Discrete Applied Mathematics - Special issue: Discrete mathematics & data mining II (DM & DM II)
Description-meet compatible multiway dissimilarities
Discrete Applied Mathematics
Description-meet compatible multiway dissimilarities
Discrete Applied Mathematics
Sub-dominant theory in numerical taxonomy
Discrete Applied Mathematics - Special issue: Discrete mathematics & data mining II (DM & DM II)
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Several approaches have been proposed for the purpose of proving that different classes of dissimilarities (e.g. ultrametrics) can be represented by certain types of stratified clusterings which are easily visualized (e.g. indexed hierarchies). These approaches differ in the choice of the clusters that are used to represent a dissimilarity coefficient. More precisely, the clusters may be defined as the maximal linked subsets, also called ML-sets; equally they may be defined as a particular type of 2-ball. In this paper, we first introduce the notion of a k-ball, thereby extending the notion of a 2-ball. For an arbitrary dissimilarity coefficient, we establish some properties of the k-balls that pinpoint the connection between them and the ML-sets. We also introduce the (2,k)-point condition (k ≥ 1) which is an extension of the Bandelt four-point condition.For k ≥ 2, we prove that the dissimilarities satisfying the (2, k)-point condition are in one-one correspondence with a class of stratified clusterings, called k-weak hierarchical representations, whose main characteristic is that the intersection of (k + 1) arbitrary clusters may be reduced to the intersection of some k of these clusters.