# Embedding Planar Graphs at Fixed Vertex Locations

## Abstract

Let *G* be a planar graph of *n* vertices, *v* _{1},..., *v* _{ n }, and let *p* _{1},... ,*p* _{ n } be a set of *n* points in the plane. We present an algorithm for constructing in *O*(*n* _{2}) time a planar embedding of *G*, where vertex *v* _{ i } is represented by point *p* _{ i } and each edge is represented by a polygonal curve with *O*(*n*) bends (internal vertices.) This bound is asymptotically optimal in the worst case. In fact, if *G* is a planar graph containing at least *m* pairwise independent edges and the vertices of *G* are randomly assigned to points in convex position, then, almost surely, every planar embedding of *G* mapping vertices to their assigned points and edges to polygonal curves has at least *m*/20 edges represented by curves with at least *m*/40^{3} bends.

## Keywords

Line Segment Planar Graph Hamiltonian Cycle Internal Vertex Outerplanar Graph## References

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