Generating Outerplanar Graphs Uniformly at Random

Generating Random Outerplanar Graphs

Asymptotic Study of Subcritical Graph Classes

We present a unified general method for the asymptotic study of graphs from the so-called “subcritical” graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from ...

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...

Every Property of Outerplanar Graphs is Testable

A D-disc around a vertex v of a graph G = (V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most n edges, by a set of D-discs around the vertices, for D = D( ) that is independent of the size of the graph. Such a result was already known for ...

Vertices of given degree in series-parallel graphs

We show that the number of vertices of a given degree k in several kinds of series-parallel labelled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labelled planar graphs. The proof method is based on generating functions and sin...