We show in this paper a bijection between totally balanced hypergraphs and so called totally balanced dissimilarities. We give an efficient way (O(n 3 ) where n is the number of elements) to (i) recognize if a given dissimilarity is totally balanced and (ii) approximate it if it is not the case. We also introduce a new kind of dissimilarity which generalize chordal graphs and allows a polynomial number of cluster that can be easily computed and interpreted.

A hypergraph H is (i) Totally balanced if it does not contain a special cycle, (ii) Binary if it is closed under intersection and every hyperedge has at most two predecessors (for inclusion order). We show in this paper that a hypergraph H is totally balanced if and only if it can be embedded into a binary hypergraph H ; H is said to be a binary extension of H. We give an efficient algorithm which, given a totally balanced hypergraph H, produces a minimal binary extension H of H; in addition, if H is a hierarchy or an interval hypergraph, then so is H.