On the geodetic iteration number of distance-hereditary graphs

For each positive integer k, we give a forbidden induced subgraph characterization of distance-hereditary graphs with geodetic iteration number at most k. Distance-hereditary graphs play an important role in the study of geodesic convexity, since for such graphs the distances between vertices are preserved in connected induced subgraphs. As a consequence of our results, we describe a polynomial-time algorithm for the computation of gin(G) when G is a distance-hereditary graph.