Graph classes with given 3-connected components: asymptotic counting, limit laws and critical phenomena

Consider a family T of 3-connected graphs, and let G be the class of graphs whose 3-connected components are graphs in T . We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions. This generalizes previously studied cases such as planar graphs and seriesparallel graphs. We provide a general theorem for the asymptotic number of graphs in G, based on the singularities of the exponential generating function associated to T . We derive limit laws for the number of connected components, for the number of edges and for the number of 2-connected components. At last, for some classes under study we show the existence of critical phenomena as the edge density in the class varies.