Binary set systems and totally balanced hypergraphs

Abstract

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.