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

Monday 7th April 2008 - 16:30 to 17:15
INI Seminar Room 1

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.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons