An Exact Enumeration of Distance-Hereditary Graphs
Cédric Chauve, Éric Fusy, Jérémie Lumbroso
备注：
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."
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 ${(7.24975\ldots)^n}$---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.
In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2…
In this paper, we build on recent results by Chauve et al. (2014) and Bahrani and Lumbroso (2017), which combined the split-decomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphs---in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic). Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs, which we describe in more detail further down in this paper…
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Hereditary properties of combinatorial structures: posets and oriented graphs
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Enumeration of symmetric centered rhombus tilings of a hexagon
一个六边形的菱形镶嵌被称为居中，如果它包含中心菱形。我们计算了边长为$a, b, a, a, b, a$且居中的六边形的垂直对称菱形镶嵌的个数。当$a$为奇数而$b$为偶数时，这表明$a, b, a, a, b, a$六边形的随机垂直对称菱形平铺居中的概率与$a, b, a, a, b, a$六边形的随机菱形平铺居中的概率完全相同。这也导致了一个因式分解定理的数量的所有菱形镶嵌的六边形是中心。
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Intractable enumeration problems are like Russian nesting dolls: structural properties of monomer-dimer coverings on two-dimensional quadratic lattices
