A diffeomorphism f of the disk has a periodic orbit [Oscr ], and the paper considers the minimal topological entropy l(f, [Oscr ]) in the isotopy class of f with respect to this orbit. The structure of the periodic orbits with l(f, [Oscr ]) = 0 has been studied lately and is now well understood. For l(f, [Oscr ]) non-zero, the paper obtains a lower bound of l(f, [Oscr ]) that depends only on the period of [Oscr ] and which improves previously known estimates. This result is a consequence of a stronger bound in the case where the isotopy class is pseudo-Anosov. It is obtained by studying the induced dynamics on a graph.