Graph classes with given 3-connected components: asymptotic counting and critical phenomena
Seminar Room 1, Newton Institute
Fix a family T of 3-connected labelled graphs with moderate growth, and let G be the class of graphs whose 3-connected 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. There are two main regimes, that we call tree-like and map-like. In the tree-like case the largest 2-connected and 3-connected cores are almost surely of constant size, whereas in the map-like case they are of linear size. For some of the classes under study we show the presence of critical phenomena as the edge density in the class varies.
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