Embedding Planar Graphs at Fixed Vertex Locations
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.
KeywordsLine Segment Planar Graph Hamiltonian Cycle Internal Vertex Outerplanar Graph
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