Enumerating topological $(n_k)$-configurations

Abstract

International audienceAn $(n_k)$-configuration is a set of $n$ points and $n$ lines in the projective plane such that the point-line incidence graph is $k$-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines or just combinatorial lines. We provide an algorithm for generating all combina- torial $(n_k)$-configurations that admit a topological realization, for given $n$ and $k$. This is done without enumerating first all combinatorial $(n_k)$-configurations. Among other results, our algorithm enables us to con- firm, in just one hour with a Java code of the second author, a satisfiability result of Lars Schewe, obtained after several months of CPU-time