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!



A proper definition of connectivity for hypergraphs
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). 


