Simple $k$-Planar Graphs are Simple $(k+1)$-Quasiplanar

TL;DR

This paper demonstrates that for any $k \\geq 2$, a simple $k$-planar graph can be transformed into a simple $(k+1)$-quasiplanar graph, establishing a relationship between these two graph classes.

Contribution

It proves that simple $k$-planar graphs are transformable into simple $(k+1)$-quasiplanar graphs for all $k \\geq 2$, revealing a new connection between these graph types.

Findings

Every simple $k$-planar graph can be converted into a $(k+1)$-quasiplanar graph.

The transformation preserves simplicity and topological properties.

Establishes a hierarchy between $k$-planar and $k$-quasiplanar graphs.

Abstract

A simple topological graph is $k$-quasiplanar ($k\geq 2$) if it contains no $k$ pairwise crossing edges, and $k$-planar if no edge is crossed more than $k$ times. In this paper, we explore the relationship between $k$-planarity and $k$-quasiplanarity to show that, for $k \geq 2$, every $k$-planar simple topological graph can be transformed into a $(k+1)$-quasiplanar simple topological graph.