Simple Topological Drawings of $k$-Planar Graphs

TL;DR

This paper proves that for any k-planar graph, there exists a simple topological drawing with a bounded number of crossings per edge, and provides an algorithm for 4-planar graphs to have an 8-plane simple drawing.

Contribution

It establishes the existence of a function bounding the crossings per edge for simple drawings of k-planar graphs and presents an algorithm for 4-planar graphs.

Findings

Existence of a function f(k) for simple drawings of k-planar graphs.

Every 4-planar graph admits an 8-plane simple drawing.

Answer to Schaefer's question on simple drawings of k-planar graphs.

Abstract

Every finite graph admits a \emph{simple (topological) drawing}, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, \emph{$k$-planar graphs} are those graphs that can be drawn so that every edge has at most $k$ crossings (i.e., they admit a \emph{$k$-plane drawing}). It is known that for $k\le 3$, every $k$-planar graph admits a $k$-plane simple drawing. But for $k\ge 4$, there exist $k$-planar graphs that do not admit a $k$-plane simple drawing. Answering a question by Schaefer, we show that there exists a function $f : \mathbb{N}\rightarrow\mathbb{N}$ such that every $k$-planar graph admits an $f(k)$-plane simple drawing, for all $k\in\mathbb{N}$. Note that the function $f$ depends on $k$ only and is independent of the size of the graph. Furthermore, we develop an algorithm to show that every $4$-planar graph admits an $8$-plane simple drawing.