# Testing Gap k-planarity is NP-complete

## TL;DR

This paper proves that determining whether a graph is k-planar or has a local crossing number within certain bounds is NP-complete, establishing computational hardness and exploring relationships between different crossing number measures.

## Contribution

It extends NP-completeness results to the gap version of k-planarity and analyzes the complexity of approximating local and total crossing numbers simultaneously.

## Key findings

- Deciding k-planarity is NP-complete for all k ≥ 1.
- The gap version of local crossing number decision is NP-complete.
- No (2-ε)-approximation algorithm exists for the local crossing number for any fixed k.

## Abstract

For all $k \geq 1$, we show that deciding whether a graph is $k$-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is NP-complete. In particular, given a graph with local crossing number either at most $k\ge 1$ or at least $2k$, we show that it is NP-complete to decide whether the local crossing number is at most $k$ or at least $2k$. This algorithmic lower bound proves the non-existence of a $(2-\epsilon)$-approximation algorithm for any fixed $k \ge 1$. In addition, we analyze the sometimes competing relationship between the local crossing number (maximum number of crossings per edge) and crossing number (total number of crossings) of a drawing. We present results regarding the non-existence of drawings that simultaneously approximately minimize both the local crossing number and crossing number of a graph.