  We call a graph $k$-geodetic, for some $k\geq 1$, if it is connected and
between any two vertices there are at most $k$ geodesics. It is shown that any
hyperbolic group with a $k$-geodetic Cayley graph is virtually-free.
Furthermore, in such a group the centraliser of any infinite order element is
an infinite cyclic group. These results were known previously only in the case
that $k=1$. A key tool used to develop the theorem is a new graph theoretic
result concerning ``ladder-like structures'' in a $k$-geodetic graph.
