We investigate the enumeration problems for the class of distance-hereditary graphs and related graph classes. We first give an enumeration algorithm for the class of distance-hereditary graphs using the recent framework for enumerating every non-isomorphic element in a graph class. In the algorithm, we use the vertex-incremental characterization of distance-hereditary graphs, which consists of three generation rules for adding one vertex to a distance-hereditary graph. Reducing the three generation rules, we can obtain vertex-incremental characterization of cographs and (6,2)-chordal bipartite graphs. These enumeration algorithms are slower than previously known theoretical enumeration algorithms, however, ours are easy to implement. In fact, we implemented our algorithm and obtained catalogs of these graph classes of up to 15 vertices. The class of Ptolemaic graphs is an intersection of the class of chordal graphs and the class of distance-hereditary graphs. We modify the enumeration algorithm for distance-hereditary graphs and obtain an enumeration algorithm for Ptolemaic graphs. The running time of the enumeration of Ptolemaic graphs is much improved from the previously known one, and we also enumerated the graphs up to 15 vertices. We also enumerate the class of 3-leaf power graphs by modifying the algorithm for Ptolemaic graphs. As far as the authors know, some of these graph classes had been counted, however, they had never been enumerated.