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Sampling Rooted 3-Connected Planar Graphs in Deterministic Polynomial Time


Johannsen,  Daniel
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Johannsen, D. (2006). Sampling Rooted 3-Connected Planar Graphs in Deterministic Polynomial Time. Master Thesis, Humboldt-Universität zu Berlin, Saarbrücken.

In this thesis an algorithm for sampling rooted 3-connected planar graphs (c-nets) in deterministic polynomial time is presented. The algorithm is based on a decomposition strategy for c-nets, which is formulated as a system of bijections between classes of c-nets parameterized by the number of vertices, faces, and edges on the outer face. This system is then used in three ways: First, as system of recursive equations to derive the sizes of above classes by using an algorithm based on dynamic programming. Second, as system of equations of generating functions to derive an algebraic generating function and a single parameter recursion formula for c-nets. Third, as composition scheme to sample c-nets uniformly at random with the recursive method of sampling. The thesis is based on the paper \emph{A Direct Decomposition of 3-connected Planar Graphs} by Manuel Bodirsky, Clemens Gr{\"o}pl, Mihyun Kang and the author~\cite{3connplanar} and extends it by the parameter for the number of edges and a full proof of the decomposition.