  The topology of the Bowditch boundary of a relatively hyperbolic group pair
gives information about relative splittings of the group. It is therefore
interesting to ask if there is generic behavior of this boundary. The purpose
of this article is to show that in the case of a non-abelian free group with
cyclic peripheral structure, there is not a generic case. This goal is
accomplished by showing the minimal size of cut sets in the boundary increases
as the lengths of the random words generating the peripheral structure go to
infinity.
