  In their paper titled "On $F$-inverse covers of inverse monoids", Auinger and
Szendrei have shown that every finite inverse monoid has an $F$-inverse cover
if and only if each finite graph admits a locally finite group variety with a
certain property. We study this property and prove that the class of graphs for
which a given group variety has the required property is closed downwards in
the minor ordering, and can therefore be described by forbidden minors. We find
these forbidden minors for all varieties of Abelian groups, thus describing the
graphs for which such a group variety satisfies the above mentioned condition.
