On the generation of topological (nk)-configurations

An (nk)-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 combinatorial (nk)-configurations that admit a topological realization, for given n and k. This is done without enumerating first all combinatorial (nk)-configurations. Among other results, our algorithm enables us to confirm, in just one hour with a Java code of the second author, a satisfiability result of Lars Schewe in [11], obtained after several months of CPU-time.