Abstract
We present polynomial-time algorithms for computing an embedding of a planar graph that minimizes the outerplanarity, or the width, or the radius, or some other measures of distance to the outer face. On the other hand, we show it is NP-hard to compute an embedding that minimizes the diameter of the dual graph.
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References
A. Aggarwal, M. Klawe, D. Lichtenstein, N. Linial, and A. Wigderson, Multi-layer grid embeddings,Proc. 26th IEEE Symp. Found. Comput. Sci, pp. 186–196, 1985.
B. Baker, Approximation algorithms for NP-complete problems in planar graphs,Proc. 24th IEEE Symp. Found. Comput. Sci, pp. 265–273, 1983.
D. Bienstock and C. L. Monma, On the complexity of covering vertices by faces in a planar graph,SIAM J. Comput.,17, pp. 53–76, 1988.
D. Bienstock and C. L. Monma, Optimal enclosing regions in a planar graph,Networks, to appear.
D. Dolev, T. Leighton, and H. Trickey, Planar embedding of planar graphs,Adv. in Comput. Res.,2, pp. 147–161, 1984.
M. R. Garey and D. S. Johnson,Computers and Intractability, Freeman, San Francisco, 1979.
L. Lovász,Combinatorial Problems and Exercises, North Holland, Amsterdam, 1979.
N. Robertson and P. D. Seymour, Graph minors. III. Planar tree-width,J. Combin. Theory Ser. B,36, 1, pp. 49–64, 1984.
R. E. Tarjan, Dividing a graph into triconnected components,SIAM J. Comput., pp. 135–158, 1973.
D. J. A. Welsh,Matroid Theory, Academic Press, New York, 1976.
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Communicated by F. Thomson Leighton.
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Bienstock, D., Monma, C.L. On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5, 93–109 (1990). https://doi.org/10.1007/BF01840379
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DOI: https://doi.org/10.1007/BF01840379