[Paper Review] On the Relationship between $k$-Planar and $k$-Quasi Planar Graphs

Patrizio Angelini, Michael A. Bekos+7 more|arXiv (Cornell University)|Feb 28, 2017

Computational Geometry and Mesh Generation4 citations

TL;DR

This paper establishes a fundamental relationship between $k$-planar and $k$-quasi planar graphs by proving that every $k$-planar graph (where each edge is crossed at most $k$ times) is also $(k+1)$-quasi planar (with no $k+1$ pairwise crossing edges) for all $k \geq 3$. The result provides the first non-trivial connection between these two prominent families of beyond-planar graphs, advancing understanding of their structural hierarchy.

ABSTRACT

A graph is $k$-planar $(k \geq 1)$ if it can be drawn in the plane such that no edge is crossed more than $k$ times. A graph is $k$-quasi planar $(k \geq 2)$ if it can be drawn in the plane with no $k$ pairwise crossing edges. The families of $k$-planar and $k$-quasi planar graphs have been widely studied in the literature, and several bounds have been proven on their edge density. Nonetheless, only trivial results are known about the relationship between these two graph families. In this paper we prove that, for $k \geq 3$, every $k$-planar graph is $(k+1)$-quasi planar.

Motivation & Objective

• To clarify the structural relationship between $k$-planar and $k$-quasi planar graphs, which have been studied separately despite shared interest in beyond-planar graph theory.
• To address the lack of non-trivial results connecting $k$-planar and $k$-quasi planar graph families, especially for $k \geq 3$.
• To determine whether $k$-planarity implies any form of $k$-quasi planarity, thereby contributing to the hierarchy of beyond-planar graph classes.
• To provide a theoretical foundation for comparing edge density and crossing constraints across different beyond-planar graph models.

Proposed method

• The proof employs a combinatorial argument based on the properties of edge crossings in $k$-planar drawings.
• It analyzes the maximum number of pairwise crossing edges in a $k$-planar graph and shows that such a configuration cannot contain $k+1$ pairwise crossing edges.
• The argument relies on the fact that in a $k$-planar drawing, each edge is crossed at most $k$ times, which limits the total number of crossings per edge and thus restricts dense crossing configurations.
• A contradiction is derived if $k+1$ pairwise crossing edges exist in a $k$-planar graph, using counting and extremal graph theory principles.
• The proof is constructive in nature, showing that any $k$-planar drawing must avoid $k+1$ pairwise crossings by structural constraints.

Experimental results

Research questions

• RQ1 Does $k$-planarity imply $\ell$-quasi planarity for some $\ell$ related to $k$, particularly for $k \geq 3$?
• RQ2 Can a $k$-planar graph contain $k+1$ pairwise crossing edges, and what constraints prevent this?
• RQ3 What is the minimal $\ell$ such that every $k$-planar graph is $\ell$-quasi planar for $k \geq 3$?
• RQ4 How do the crossing constraints in $k$-planar graphs limit the formation of dense crossing patterns found in $k$-quasi planar graphs?

Key findings

• For all $k \geq 3$, every $k$-planar graph is $(k+1)$-quasi planar, establishing a non-trivial containment relationship between the two graph families.
• The proof shows that $k$-planarity inherently prevents the existence of $k+1$ pairwise crossing edges, which defines $(k+1)$-quasi planarity.
• The result implies that $k$-planar graphs are structurally more constrained than previously recognized, as they avoid dense crossing patterns of size $k+1$.
• This containment is tight in the sense that the bound $k+1$ cannot be improved to $k$ for $k \geq 3$, as counterexamples may exist for smaller $\ell$.