A characterization of totally balanced hypergraphs

AbstractA hypergraph is totally balanced if every non-trivial cycle has an edge containing at least three vertices of the cycle. Totally balanced hypergraphs are characterized here as special tree-hypergraphs. This approach provides a conceptually simpler proof of Anstee's related result and yields the structural description of totally balanced hypergraphs

Some characterizations of gamma and beta-acyclicity of hypergraphs

The notions of $\gamma$ and $\beta$-acyclicity are two classic generalizations of the acyclicity of graphs to hypergraphs. They satisfy the property that, if a hypergraph is $\gamma$-acyclic then it is $\beta$-acyclic, and the reverse is false. We give some new properties concerning these notions. First we show that we can strictly insert another notion of acyclicity between them, namely the fact of having a join tree with disjoint branches. And if we add a condition on the existence of such a join tree, we obtain a notion equivalent to $\gamma$-acyclicity. Then we present two characterizations, consisting in applying successively a small set of rules, deciding $\gamma$ and $\beta$-acyclicity respectively

Workshop Notes of the Sixth International Workshop "What can FCA do for Artificial Intelligence?"

International audienc