# Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

## TL;DR

This paper introduces a systematic method to generate all non-isomorphic topological representations of complete and bipartite graphs within beyond-planarity classes, leading to new bounds on their properties.

## Contribution

The paper presents a novel technique for generating all non-isomorphic representations of certain graphs under beyond-planarity constraints, surpassing traditional combinatorial methods.

## Key findings

- Generated all non-isomorphic topological representations for selected graph classes.
- Achieved new tight bounds for maximum edge density in beyond-planarity classes.
- Demonstrated the technique's effectiveness across various graph classes.

## Abstract

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.