Saturated $2$-planar drawings with few edges
J\'anos Bar\'at, G\'eza T\'oth

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 -plane if every edge contains at most crossings. A -plane drawing is saturated if we cannot add any edge so that the drawing remains -plane. It is well-known that saturated -plane drawings, that is, maximal plane graphs, of vertices have exactly edges. For , the number of edges of saturated -vertex -plane graphs can take many different values. In this note, we establish some bounds on the minimum number of edges of saturated -plane graphs under different conditions. If two edges can cross at most once, then such a graph has at least edges. If two edges can cross many times, then we show the tight bound of for the number of edges.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Digital Image Processing Techniques · Advanced Graph Theory Research
