  Carrier graphs were first introduced for closed hyperbolic 3-manifolds by
White. In this paper, we first generalize this definition to carrier graphs for
representations of a rank two free group into the isometry group of hyperbolic
three space. Then we prove the existence and the finiteness of minimal carrier
graphs for those representations which are discrete, faithful and geometrically
finite, and more generally, those that satisfy certain finiteness conditions
first introduced by Bowditch.
