  Beyond-planarity focuses on combinatorial properties of classes of non-planar
graphs that allow for representations satisfying certain local geometric or
topological constraints on their edge crossings. Beside the study of a specific
graph class for its maximum edge density, another parameter that is often
considered in the literature is the size of the largest complete or complete
bipartite graph belonging to it.
  Overcoming the limitations of standard combinatorial arguments, we present a
technique to systematically generate all non-isomorphic topological
representations of complete and complete bipartite graphs, taking into account
the constraints of the specific class. As a proof of concept, we apply our
technique to various beyond-planarity classes and achieve new tight bounds for
the aforementioned parameter.
