An Experimental Study of a 1-planarity Testing and Embedding Algorithm

TL;DR

This paper explores a backtracking-based algorithm for testing and embedding 1-planar graphs, demonstrating feasibility on small graphs and highlighting the need for more advanced methods for larger instances.

Contribution

It presents the first implementation of a general 1-planarity testing and embedding algorithm using backtracking, with experimental results on small graphs.

Findings

Successful application to graphs with up to 30 vertices

Indicates the need for more sophisticated techniques for larger graphs

Provides initial insights for practical 1-planarity testing approaches

Abstract

The definition of $1$-planar graphs naturally extends graph planarity, namely a graph is $1$-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time, deciding whether a graph is $1$-planar is NP-complete, even for restricted classes of graphs. Although several polynomial-time algorithms have been described for recognizing specific subfamilies of $1$-planar graphs, no implementations of general algorithms are available to date. We investigate the feasibility of a $1$-planarity testing and embedding algorithm based on a backtracking strategy. While the experiments show that our approach can be successfully applied to graphs with up to 30 vertices, they also suggest the need of more sophisticated techniques to attack larger graphs. Our contribution provides initial indications that may stimulate further research on the design of practical approaches for the $1$-planarity testing problem.