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

Abstract

Fix a family T of 3-connected graphs, and let G be the class of graphs whose 3connected components are the graphs in T . We present a general framework for analyzing such graph classes based on singularity analysis of generating functions. This generalizes previously studied cases such as planar graphs and series-parallel 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 . For some of the classes under study we show the presence of critical phenomena as the edge density in the class varies.