Binary set systems and totally balanced hypergraphs
Résumé
A hypergraph $H$ is
{\it(i) Totally balanced} if it does not contain a special cycle,
{\it(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 {\it binary extension} of $H$.
We give an efficient algorithm which, given a totally balanced hypergraph $H$, produces a minimal binary extension $\widehat{H}$ of $H$;
in addition, if $H$ is a hierarchy or an interval hypergraph, then so is $\widehat{H}$.
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