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Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on n vertices is bounded by 2^ 3.59n. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providingarXiv:1608.01464v1 fatcat:lmepfq33rvd7pd3vpjj3d2r5uy