Saturated $2$-planar drawings with few edges

TL;DR

This paper investigates the minimum number of edges in saturated 2-plane graph drawings, establishing bounds based on crossing conditions, and contributes to understanding their structural properties.

Contribution

It provides new bounds on the minimum edges in saturated 2-plane graphs under different crossing constraints.

Findings

Graphs with at most one crossing per pair of edges have at least n-1 edges.

Graphs allowing multiple crossings per pair have a minimum of approximately 2n/3 edges.

The paper establishes tight bounds for these classes of saturated 2-plane graphs.

Abstract

A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is, maximal plane graphs, of $n$ vertices have exactly $3n-6$ edges. For $k>0$, the number of edges of saturated $n$-vertex $k$-plane graphs can take many different values. In this note, we establish some bounds on the minimum number of edges of saturated $2$-plane graphs under different conditions. If two edges can cross at most once, then such a graph has at least $n-1$ edges. If two edges can cross many times, then we show the tight bound of $\lfloor2n/3\rfloor$ for the number of edges.