Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

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.

The Number of Edges in Maximal 2-planar Graphs

“…So in each iteration we have a partial drawing Γ and a collection H of vertices and edges still to be drawn. We then exhaustively explore the space of simple 2-plane drawings of Γ ∪ H. Our approach is similar to the one used by Angelini, Bekos, Kaufmann, and Schneck [2] for complete and complete bipartite graphs. We consider the edges to be drawn in some order such that whenever an edge is considered, at least one of its endpoints is in the drawing already.…”

“…We prove Lemma 31 using an exhaustive computational exploration of the space of simple drawings for X. Our approach is similar to the one used by Angelini, Bekos, Kaufmann, and Schneck [2] for topological drawings of complete and complete bipartite graphs. The drawings are built incrementally and represented as a doubly-connected edge list (DCEL) [7,Chapter 2].…”