Drawing Planar Graphs of Bounded Degree with Few Slopes

We settle a problem of Dujmovic, 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 one bend, then $2d$ slopes suffice. Allowing 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. Every other planar graph requires exactly $\lceil d/2\rceil$ slopes.

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