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arXiv · 1009.1315

Drawing planar graphs of bounded degree with few slopes

Balázs Keszegh·János Pach·Dömötör Pálvölgyi
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Abstract

We settle a problem of Dujmovi\'c, Eppstein, Suderman, and Wood by showing that there exists a function $f$ with the property that every planar graph $G$ with maximum degree $d$ admits a drawing with noncrossing straight-line edges, using at most $f(d)$ different slopes. If we allow the edges to be represented by polygonal paths with {\em one} bend, then 2d slopes suffice. Allowing {\em two} bends per edge, every planar graph with maximum degree $d\ge 3$ can be drawn using segments of at most $\lceil d/2\rceil$ different slopes. There is only one exception: the graph formed by the edges of an octahedron is 4-regular, yet it requires 3 slopes. These bounds cannot be improved.

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Balázs Keszegh, János Pach, Dömötör Pálvölgyi. 2010-09-07. Drawing planar graphs of bounded degree with few slopes. https://arxiv.org/abs/1009.1315

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