An Exact Enumeration of Distance-Hereditary Graphs [article]

Cédric Chauve and Éric Fusy and Jérémie Lumbroso
2016 arXiv   pre-print
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 providing
more » ... an exact enumeration of distance-hereditary graphs, which allows to show that the number of distance-hereditary graphs on n vertices is tightly bounded by (7.24975...)^n---opening the perspective such graphs could be encoded on 3n bits. We also provide the exact enumeration and asymptotics of an important subclass, the 3-leaf power graphs. Our work illustrates the power of revisiting graph decomposition results through the framework of analytic combinatorics.
arXiv:1608.01464v1 fatcat:lmepfq33rvd7pd3vpjj3d2r5uy