An Exact Enumeration of Distance-Hereditary Graphs

Cédric Chauve, Éric Fusy, Jérémie Lumbroso
2017 2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
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 distancehereditary 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 » ... n exact enumeration and full asymptotic of distance-hereditary graphs, which allows to show that the number of distancehereditary 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 full asymptoticss 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. *
doi:10.1137/1.9781611974775.3 dblp:conf/analco/ChauveFL17 fatcat:cum2zdrtrnfwdo7qdbs3lh5rgq