, Volume 17, Issue 4, pp 717-728

Embedding Planar Graphs at Fixed Vertex Locations

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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/403 bends.

Received: May 24, 1999 Final version received: April 10, 2000