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A transversal of a hypergraph H is a subset of vertices that
intersects all the hyper-edges H. The enumeration and the counting of
the minimal transversals have a lot of applications in many domains
(graph theory, AI, datamining, etc). Counting problems are generally
harder than theirs associated decision problems. For example, finding
a minimal transversal is doable in polynomial time but counting them
is #P-complet (the equivalent of NP-complet for counting problems).

We have proved that we can count the minimal transversals of any
beta-acyclique hypergraph in polynomial time. Our result is based on
a recursive decomposition of the beta-acyclique hypergraph founded by
Florent Capelli and by the introduction of a new notion that
generalize the minimal transversals.

A lot of exciting open questions live in the neighborhood of our
result. In particular, our algorithm is able to count the minimum
dominating set of a strong-chordal graph. But counting the minimum
dominating set is #P-complete on split graphs. Is it the beginning of
a complete characterization of the complexity of counting minimal
dominating sets in dense graphs ?
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