Given a graph class $\mathcal G$, let ${\mathcal G}_n$ be the set of graphs of $\mathcal G$ on the vertex set $.[n]$. For a given class $\mathcal G$, we are interested in the asymptotic behavior of random graphs $R_n$ uniformly sampled from ${\mathcal G}_n$. If $n |{\mathcal G}_{n-1}| then call $\mathcal G$ smoothly. / |{\mathcal G}_n|$ tends to be limited as $n \to \infty$. Showing that the graph class is smooth involves investigating the properties of $R_n$, especially the asymptotic probabilities that $R_n$ is connected, and more generally the asymptotic behavior of fragments of $R_n$ other than the maximum value. is an important step in the approach to component.
Bender’s, Canfield’s, and Richmond’s construction methods show that a class of graphs that can be embedded in a particular surface is smooth. Similarly, there is minor closed class smoothness for graphs with 2-connected excluded minors. Here we develop our approach further and present results that include both these and other cases. Under very general conditions, the graph class is smooth, allowing us to describe, for example, the restricted distribution of fragments of $R_n$ and the size of the core. We also get similar results for graphs of classes with a minimum degree of at least 2.
