Adding one edge to planar graphs makes crossing number and 1-planarity hard

TL;DR

This paper proves that adding a single edge to a planar graph makes computing the crossing number and determining 1-planarity NP-hard, highlighting the computational complexity of near-planar graphs.

Contribution

It establishes NP-hardness results for crossing number and 1-planarity in near-planar graphs, introduces anchored embedding, and provides a new geometric proof for cubic graphs.

Findings

NP-hardness of crossing number for near-planar graphs

NP-hardness of 1-planarity decision for near-planar graphs

New geometric proof of NP-completeness for crossing number in cubic graphs

Abstract

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hlin\v{e}n\'y.