Enumeration of Unlabeled Outerplanar Graphs

TL;DR

This paper provides exact and asymptotic counts of unlabeled outerplanar graphs, along with statistical properties, using advanced combinatorial and analytic techniques.

Contribution

It introduces a polynomial-time method for counting unlabeled outerplanar graphs and derives their asymptotic enumeration formula.

Findings

Exact count of unlabeled outerplanar graphs computed in polynomial time

Asymptotic formula for the number of such graphs derived

Statistical properties like connectivity and chromatic number analyzed

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 $g n^{-5/2}\rho^{-n}$, where $g\approx0.00909941$ and $\rho^{-1}\approx7.50360$ 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.