Min-$k$-planar Drawings of Graphs

TL;DR

This paper introduces min-$k$-planar graph drawings, generalizing $k$-planar drawings by allowing edges to cross arbitrarily but with restrictions on crossing pairs, and establishes bounds on the number of edges and inclusion relations.

Contribution

It defines min-$k$-planar drawings, provides upper bounds on edge counts, and explores their relationship with $k$-planar graphs.

Findings

Established a general upper bound on edges for min-$k$-planar drawings.

Derived a tighter bound for the case $k=3$.

Proved tight bounds for $k=1,2$.

Abstract

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the $k$-planar drawings $(k \geq 1)$, where each edge cannot cross more than $k$ times. We generalize $k$-planar drawings, by introducing the new family of min-$k$-planar drawings. In a min-$k$-planar drawing edges can cross an arbitrary number of times, but for any two crossing edges, one of the two must have no more than $k$ crossings. We prove a general upper bound on the number of edges of min-$k$-planar drawings, a finer upper bound for $k=3$, and tight upper bounds for $k=1,2$. Also, we study the inclusion relations between min-$k$-planar graphs (i.e., graphs admitting min-$k$-planar drawings) and $k$-planar graphs. In our setting we only allow simple drawings, that is, any two edges cross at most once, no two adjacent edges cross, and no three edges intersect at a common crossing point.