Let $f(G)$ be the smallest $m$, such that one can find $2m$ vertices in $G$ with the following property: pair up the vertices in any way, and find $m$ paths that join each pair. Then every set of path constructed will have a intersection. (Is there a name for this graph property?)
I have a feeling this property might be connected to the connectedness of a graph. but for large grid graphs, I can chose many vertices and still find disjoint paths connecting the pairs, when it is only 4 connected.
For a $n\times n$ grid graph $G$, what is $f(G)$?
Edit: I have found the name of the property. A graph is $k$-linked if there exist $2k$ elements, such that there are $k$ disjoint paths that pairs two of them.