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arXiv · 2110.12781

Saturated $2$-planar drawings with few edges

János Barát·Géza Tóth
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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.

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János Barát, Géza Tóth. 2021-10-25. Saturated $2$-planar drawings with few edges. https://doi.org/10.26493/1855-3974.2805.b49

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