Drawing planar graphs of bounded degree with few slopes

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.