Recognizing and Embedding Simple Optimal 2-Planar Graphs

TL;DR

This paper presents a linear-time algorithm for recognizing and embedding simple optimal 2-planar graphs, advancing understanding of their structure and computational complexity within beyond-planar graph classes.

Contribution

It introduces a combinatorial characterization and a linear-time recognition and embedding algorithm for simple optimal 2-planar graphs, filling a gap in the graph recognition literature.

Findings

Recognition of simple optimal 2-planar graphs is achievable in linear time.

A combinatorial characterization of these graphs is established.

The recognition problem for these graphs is solvable efficiently.

Abstract

In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs, the recognition problem has been settled, namely it is NP-complete for the general case, while optimal 1-planar graphs, i.e. those with maximum density, can be recognized in linear time. For 2-planar graphs, the picture is less complete. As expected, the recognition problem has been found to be NP-complete in general. In this paper, we consider the recognition of simple optimal 2-planar graphs. We exploit a combinatorial characterization of such graphs and present a linear time algorithm for recognition and embedding.