On the Degree Sequences of Random Outerplanar and Series-Parallel Graphs

In order to perform an average-case analysis for specific input distributions one needs to derive and understand properties of a ’typical’ input instance. In the case of the uniform distribution on the class of all graphs on a given vertex set, a ’typical’ input instance can be viewed as a random graph in the Erdős-Renyi model – which was intensively studied over the last decades. The situation changes if we consider graph classes which have structural side constraints. A typical example in this context is the class of planar graphs: in a random planar graph Rn (a graph drawn uniformly at random from the class of all labeled planar graphs on n vertices) the edges are obviously far from being independent. Consequently, so far basically all results about properties of random graphs with structural side constraints rely on completely different methods, mostly from analytic combinatorics. In other words, properties were obtained by precisely counting the objects in question, not using any of the sophisticated tools from random graph theory that were developed over the last years. In this paper we show that combining approaches from modern asymptotic enumeration, cf. the forthcoming book by Flajolet and Sedgewick [9], and the recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer [8] and Fusy [11] allows us to exploit standard tools from modern probability theory in order to obtain properties of typical instances of the graph classes in question. We elaborate our ideas by determining the degree sequence of a random object, a property that seemed hard to tackle by just using methods from analytic combinatorics. In the first part of the paper we develop a general framework that allows us to mechanically derive the degree distribution of random graphs from certain ’nice’ classes of graphs from the degree distribution of the 2-connected objects in these classes. In the second part of the paper we then use this framework to obtain the degree distribution of a random outerplanar graph and a random series-parallel graph. For the latter we formulate a generic concentration result that allows us to make high probability statements for a large family of variables defined on random graphs drawn according to the Boltzmann model.