  In the theory of surface diffeomorphisms relative to homoclinic and
heteroclinic orbits, it is possible to compute a one-dimensional representative
map for any irreducible isotopy class. The topological entropy of this graph
representative is equal to the growth rate of the number of essential Nielsen
classes of a given period, and hence is a lower bound for the topological
entropy of the diffeomorphism. In this paper, we show that this entropy bound
is the infemum of the topological entropies of diffeomorphisms in the isotopy
class, and give necessary and sufficient conditions for the infemal entropy to
be a minimum.
