Edge-Minimum Saturated k-Planar Drawings

TL;DR

This paper develops a framework to determine the minimum number of edges in saturated $k$-planar graph drawings under various restrictions, revealing tight bounds for sparsity in these configurations.

Contribution

It introduces a generic method to find lower bounds on edges in saturated $k$-planar drawings across multiple classes, providing tight bounds for sparsity.

Findings

Minimum edges for saturated $k$-planar drawings with incident crossings: $rac{2}{k - (k mod 2)} (n-1)$.

Minimum edges for saturated $k$-planar drawings without crossings incident, multicrossings, or selfcrossings: $rac{2(k+1)}{k(k-1)}(n-1)$.

Framework applies to various restrictions, establishing tight bounds on sparsity.

Abstract

For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to saturated graphs in a graph class as introduced by Tur\'an (1941) and Erd\H{o}s, Hajnal, and Moon (1964). We focus on $k$-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most $k$ times, and the classes $\mathcal{D}$ of all $k$-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated $k$-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all $n$-vertex saturated $k$-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest $n$-vertex saturated $k$-planar drawings have $\frac{2}{k - (k \bmod 2)} (n-1)$ edges for any $k \geq 4$, while if all that is forbidden, the sparsest such drawings have $\frac{2(k+1)}{k(k-1)}(n-1)$ edges for any $k \geq 6$.