TITLE: A proper definition of connectivity for hypergraphs
QUESTION [7 upvotes]: For usual graphs on $n$ vertices, a edge-minimal connected graph is nothing but a spanning tree of this graph. It is well-known that any spanning tree has $n-1$ edges. 
I would like to know whether there is a similar definition of connectivity for $3$-uniform hypergraphs which preserves this property. To be more specific, is there any known definition of connectivity for $3$-uniform hypergraphs such that every edge-minimal connected graph on $n$ vertices has exactly $\binom{n-1}{2}$ edges? 
I personally have a definition for any $k$-uniform hypergraphs which satisfies this property, and I just want to know whether some definitions have already been posed before. Thanks!

REPLY [6 votes]: Think of the hypergraph as a simplicial complex $\Delta$, with the facets being the hyperedges. Consider property (*) as: 

1) The $i$-skeleton of $\Delta$ is full for $0\leq i\leq k-2$ and
  2) $\tilde H_i(\Delta)=0$ for $0\leq i\leq k-2$

Then these conditions force $f_{k-1}-\binom{n-1}{k-1}= \dim \tilde H_{k-1}(\Delta)$. 
Thus a minimal complex satisfying * would have 
$f_{k-1}=\binom{n-1}{k-1}$. Such complex would have full $i$-skeleton up to $i=k-2$ and acyclic, so a reasonable generalization of trees. Of course, they will not be "spanning" in general as not all connected complex contains one, but if you restricts to (*) complexes, perhaps they will. 
I am not an expert, but remember seeing many papers studying higher-dimensional trees, so you may find this somewhere (or people who really know this area may be able to locate a reference).