Graph classes with given 3‐connected components: Asymptotic enumeration and random graphs

Abstract

AbstractConsider a family \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{T}\end{align*}\end{document} of 3‐connected graphs of moderate growth, and let \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{G}\end{align*}\end{document} be the class of graphs whose 3‐connected components are graphs in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{T}\end{align*}\end{document}. 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 \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{G}\end{align*}\end{document}, based on the singularities of the exponential generating function associated to \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{T}\end{align*}\end{document}. 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 was already observed by Panagiotou and Steger [25], and we provide a finer description. For some classes under study both regimes occur, because of a critical phenomenon as the edge density in the class varies. Finally, we analyze the size of the largest 3‐connected component in random planar graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 438–479, 2013