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.
Supported by NSF grant CCR-94-24398, OTKA-T-020914, and by a PSC-CUNY Research Award.
Supported by NSA grant MDA904-97-1-0018 and by DIMACS.
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© 1998 Springer-Verlag Berlin Heidelberg
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Pach, J., Wenger, R. (1998). Embedding Planar Graphs at Fixed Vertex Locations. In: Whitesides, S.H. (eds) Graph Drawing. GD 1998. Lecture Notes in Computer Science, vol 1547. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-37623-2_20
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DOI: https://doi.org/10.1007/3-540-37623-2_20
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