  In this paper we consider some results obtained for graphs using minimal
vertex separators and generalized chordality and translate them to the context
of Geometric Group Theory. Using these new tools, we are able to give two new
characterizations for a group to be virtually free. Furthermore, we prove that
the Baumslag-Solitar group $BS(1,n)$ is $k$-chordal for some $k$ if and only if
$|n|<3$ and we give an application of generalized chordality to the study of
the word problem.
