[Paper Review] Enumeration of Unlabeled Outerplanar Graphs

Manuel Bodirsky, Éric Fusy+2 more|ArXiv.org|Nov 16, 2005

Advanced Combinatorial Mathematics20 references3 citations

TL;DR

This paper presents a polynomial-time algorithm to compute the exact number of unlabeled outerplanar graphs on $ n $ vertices and derives their asymptotic enumeration as $ g hinspace n^{-5/2} \rho^{-n} $, with $ g \approx 0.00909941 $ and $ \rho^{-1} \approx 7.50360 $. The approach combines cycle index enumeration with singularity analysis of generating functions, enabling asymptotic analysis of structural properties such as edge count, chromatic number, and component distribution.

ABSTRACT

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number g_n of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and g_n is asymptotically $g n^{-5/2}ρ^{-n}$, where $g\approx0.00909941$ and $ρ^{-1}\approx7.50360$ can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n vertices, for instance concerning connectedness, chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.

Motivation & Objective

• To determine the exact and asymptotic number of unlabeled outerplanar graphs on $ n $ vertices.
• To develop a polynomial-time algorithm for computing the exact count $ g_n $ of unlabeled outerplanar graphs.
• To analyze typical structural properties of random unlabeled outerplanar graphs, including connectedness, chromatic number, number of edges, and component distribution.
• To extend classical combinatorial enumeration techniques to handle symmetries in unlabeled outerplanar graphs using cycle indices and singularity analysis.

Proposed method

• Use of cycle index sums to encode symmetries in unlabeled outerplanar graphs, enabling enumeration up to isomorphism.
• Application of the singular implicit function theorem to analyze generating functions defined implicitly by multiset constructions of two-connected components.
• Employment of singularity analysis on generating functions to extract asymptotic growth rates and limit laws.
• Decomposition of outerplanar graphs into two-connected blocks and connected components to model their structure via generating functions.
• Use of analytic combinatorics tools, including the transfer theorem and Gaussian limit laws, to derive distributional properties of graph parameters.
• Numerical approximation of key constants such as $ \rho^{-1} \approx 7.50360 $ and $ g \approx 0.00909941 $ using implicit equations and derivatives.

Experimental results

Research questions

• RQ1 What is the exact number of unlabeled outerplanar graphs on $ n $ vertices, and can it be computed efficiently?
• RQ2 What is the asymptotic growth rate of the number of unlabeled outerplanar graphs as $ n \to \infty $?
• RQ3 What is the typical distribution of the number of edges in a random unlabeled outerplanar graph?
• RQ4 What is the chromatic number of a random unlabeled outerplanar graph, and how does it behave asymptotically?
• RQ5 How does the number of components and isolated vertices behave in random unlabeled outerplanar graphs?

Key findings

• The number of unlabeled outerplanar graphs on $ n $ vertices is asymptotically $ g \thinspace n^{-5/2} \rho^{-n} $, with $ g \approx 0.00909941 $ and $ \rho^{-1} \approx 7.50360 $.
• The expected number of edges in a random unlabeled outerplanar graph is asymptotically $ 1.54894n $, with variance $ 0.227504n $, following a Gaussian limit law.
• The chromatic number of a random unlabeled outerplanar graph is almost surely 3, both in the unlabeled and labeled cases.
• The probability of connectivity in a random unlabeled outerplanar graph tends to 1 as $ n \to \infty $, while the expected number of components is approximately 1.17847.
• The number of isolated vertices in a random unlabeled outerplanar graph follows a geometric distribution with parameter $ \rho \approx 0.13326 $, and the expected number is $ 0.153761 $.
• The asymptotic distribution of the number of edges in both rooted and unrooted connected outerplanar graphs is Gaussian, with mean $ \mu \approx 1.54894 $ and variance $ \sigma^2 \approx 0.227504 $.