Random Graphs from Planar and Other Addable Classes.

We study various properties of a random graph Rn, drawn uniformly at random from the class $$ \mathcal{A}_n $$ of all simple graphs on n labelled vertices that satisfy some given property, such as being planar or having tree-width at most κ. In particular, we show that if the class $$ \mathcal{A} $$ is' small' and ‘addable', then the probability that Rn is connected is bounded away from 0 and from 1. As well as connectivity we study the appearances of subgraphs, and thus also vertex degrees and the numbers of automorphisms. We see further that if $$ \mathcal{A} $$ is' smooth' then we can make much more precise statements for example concerning connectivity. [ABSTRACT FROM AUTHOR]