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Drawing 3-polytopes with good vertex resolution. (English)
J. Graph Algorithms Appl. 15, No. 1, 33-52 (2011).
Summary: We study the problem how to obtain a small drawing of a 3-polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a one-dimensional problem, since it is sufficient to guarantee distinct integer $x$-coordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained inside a $2(n - 2) \times 2 \times 1$ box. The constructed embedding can be scaled to a grid embedding whose $x$-coordinates are contained in $[0, 2(n - 2)]$. Furthermore, the point set of the embedding has a small spread, which differs from the best possible spread only by a multiplicative constant.