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In this work we study properties of random graphs that are drawn uniformly at random from the class consisting of biconnected outerplanar graphs, or equivalently dissections of large convex polygons. We obtain very sharp concentration results for the number of vertices of any given degree, and for the number of induced copies of a given fixed graph. Our method gives similar results for random graphs from the class of triangulations of convex polygons.

In this work we determine the expected number of vertices of degree k = k(n) in a graph with n vertices that is drawn uniformly at random from a subcritical graph class. Examples of such classes are outerplanar, series-parallel, cactus and clique graphs. Moreover, we provide exponentially small bounds for the probability that the quantities in question deviate from their expected values. † Parts of this work appeared as an extended abstract in N. Bernasconi, K. Panagiotou and A. Steger, 'On the degree sequences of random outerplanar and series-parallel graphs ', in APPROX-RANDOM 2008 , pp. 303-316. at https:/www.cambridge.org/core/terms. https://doi.The picture changes dramatically if we are interested in natural graph classes. A standard example that has evolved over the last decade as a reference model in this context is the class of planar graphs. The random planar graph R n was first investigated in [7] by Denise, Vasconcellos and Welsh and has attracted considerable attention since then. We mention selectively a few results. McDiarmid, Steger and Welsh [16] showed the surprising fact that R n does not share the 0-1 law known from standard random graph theory: the probability of connectedness is bounded away from 0 and 1 by positive constant values; moreover, the situation is similar if the average degree is fixed [13]. These results relied on a (crude) counting of the number of planar graphs with n vertices. A breakthrough occurred with the recent results of Giménez and Noy [14], who not only managed to determine the asymptotic value of the number of planar graphs with n vertices, but also showed that the number of edges in R n is asymptotically normally distributed. Moreover, they studied the number of connected and 2-connected components in R n . The proofs of these results are based on singularity analysis of generating functions, a powerful method from analytic combinatorics that has led to many beautiful results: see the book by Flajolet and Sedgewick [10].Our results. In this paper we further elaborate and extend significantly an approach that was used in [3] to obtain the degree sequence and subgraph counts of random dissections of convex polygons. More precisely, we exploit the so-called Boltzmann sampler framework by Duchon, Flajolet, Louchard and Schaeffer [9] to reduce the study of degree sequences to properties of sequences of independent and identically distributed random variables. Hence, we can -and do -use many tools developed in classical random graph theory to obtain extremely tight results.Our first main contribution is a general framework that allows us to derive mechanically the degree distribution of random graphs from certain 'nice' graph classes, which are 'subcritical' in a well-defined analytic sense: see Section 3 for details. Our framework can be readily applied to obtain the degree sequence of random graphs from 'simple' classes, such as Cayley treesi.e., (non-plane) labelled trees -or graphs which have the property that their maximal 2-connected components (or equivalently,...

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