  Consider a family $\mathcal{T}$ of 3-connected graphs of moderate growth, and
let $\mathcal{G}$ be the class of graphs whose 3-connected components are
graphs in $\mathcal{T}$. We present a general framework for analyzing such
graphs classes based on singularity analysis of generating functions, which
generalizes previously studied cases such as planar graphs and series-parallel
graphs. We provide a general result for the asymptotic number of graphs in
$\mathcal{G}$, based on the singularities of the exponential generating
function associated to $\mathcal{T}$. We derive limit laws, which are either
normal 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 we
find a fundamental dichotomy: classes similar to planar graphs have almost
surely a unique block of linear size, while classes similar to series-parallel
graphs have only sublinear blocks. This dichotomy also applies to the size of
the largest 3-connected component. For some classes under study both regimes
occur, because of a critical phenomenon as the edge density in the class
varies.
