2016/08/04 by Cédric Chauve, Éric Fusy, Chauve, Cédric +3
Computer Science · Mathematics · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #cs.DM #math.CO
paper · pdf · doi:10.48550/arxiv.1608.01464
14 pages, 4 figures, and a Maple worksheet; preprint of long version of IGCT 2014 article presented as "An Enumeration of Distance-Hereditary and 3-Leaf Power Graphs."
arxiv created 2016/08/04 · arxiv updated 2016/08/05
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\lceil 3.59n\rceil. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing 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.