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Consider a family $\mathcal{T}$ of 3-connected graphs of moderate growth, andlet $\mathcal{G}$ be the class of graphs whose 3-connected components aregraphs in $\mathcal{T}$. We present a general framework for analyzing suchgraphs classes based on singularity analysis of generating functions, whichgeneralizes previously studied cases such as planar graphs and series-parallelgraphs. We provide a general result for the asymptotic number of graphs in$\mathcal{G}$, based on the singularities of the exponential generatingfunction associated to $\mathcal{T}$. We derive limit laws, which are eithernormal or Poisson, for several basic parameters, including the number of edges,number of blocks and number of components. For the size of the largest block wefind a fundamental dichotomy: classes similar to planar graphs have almostsurely a unique block of linear size, while classes similar to series-parallelgraphs have only sublinear blocks. This dichotomy also applies to the size ofthe largest 3-connected component. For some classes under study both regimesoccur, because of a critical phenomenon as the edge density in the classvaries.

Graph classes with given 3-connected components: asymptotic enumeration and random graphs