Minimal obstructions for 1-immersions and hardness of 1-planarity testing

TL;DR

This paper characterizes minimal non-1-planar graphs through infinite families, establishes their abundance for large sizes, and proves that testing 1-planarity is NP-complete.

Contribution

It constructs infinite families of minimal non-1-planar graphs and proves the NP-completeness of 1-planarity testing.

Findings

Constructed two infinite families of minimal non-1-planar graphs.

Showed exponential growth of such graphs for large sizes.

Proved that testing 1-planarity is NP-complete.

Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planar for every edge $e$ of $G$. We construct two infinite families of minimal non-1-planar graphs and show that for every integer $n > 62$, there are at least $2^{(n-54)/4}$ nonisomorphic minimal non-1-planar graphs of order $n$. It is also proved that testing 1-planarity is NP-complete.