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We consider embeddings of planar graphs in $R^2$ where vertices map to pointsand edges map to polylines. We refer to such an embedding as a polylinedrawing, and ask how few bends are required to form such a drawing for anarbitrary planar graph. It has long been known that even when the vertexlocations are completely fixed, a planar graph admits a polyline drawing whereedges bend a total of $O(n^2)$ times. Our results show that this number ofbends is optimal. In particular, we show that $\Omega(n^2)$ total bends isrequired to form a polyline drawing on any set of fixed vertex locations foralmost all planar graphs. This result generalizes all previously known lowerbounds, which only applied to convex point sets, and settles 2 open problems.

The Minimum Bends in a Polyline Drawing with Fixed Vertex Locations