Zhi-Zhong Chen
Abstract
We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, derive some consequences regarding sparsity and coloring, and survey some algorithmic results.
- Allen, J. F. 1983. Maintaining knowledge about temporal intervals. Commun. ACM 26, 11 (Nov.), 832--843.]] Google Scholar
- Borodin, O. V. 1984. Solution of Ringel's problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs. Metody Diskret. Analiz. 41, 12--26. In Russian.]]Google Scholar
- Borodin, O. V. 1992. Cyclic coloring of plane graphs. Discrete Math. 100, 281--289.]] Google Scholar
- Borodin, O. V. 1995. A new proof of the 6 color theorem. J. Graph Theory 19, 4, 507--522.]] Google Scholar
- Chen, Z.-Z. 2001. Approximation algorithms for independent sets in map graphs. J. Algorithms 41, 1, 20--40.]] Google Scholar
- Chen, Z.-Z., Grigni, M., and Papadimitriou, C. 1998. Planar map graphs. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC). ACM, New York, pp. 514--523.]] Google Scholar
- Chen, Z.-Z., Grigni, M., and Papadimitriou, C. 1999. Map graphs. Manuscript, arXiv:cs.DM/ 9910013, 46 pages.]]Google Scholar
- Chen, Z.-Z., and He, X. 2000. Hierarchical topological inference on planar disc maps. In Proceedings of the 6th International Computing and Combinatorics Conference (COCOON). Springer-Verlag, Berlin, Germany, pp. 115--125.]] Google Scholar
- Chen, Z.-Z., He, X., and Kao, M.-Y. 1999b. Nonplanar topological inference and political-map graphs. In Proceedings of the 10th Annual ACM---SIAM Symposium on Discrete Algorithms (SODA). ACM, New York, pp. 195--204.]] Google Scholar
- Ehrlich, G., Even, S., and Tarjan, R. E. 1976. Intersection graphs of curves in the plane. J. Combinat. Theory Ser. B 21, 1, 8--20.]]Google Scholar
- Grigni, M., Papadias, D., and Papadimitriou, C. 1995. Topological inference. In Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI). IJCAI, Inc., Somerset, N.J., pp. 901--907.]]Google Scholar
- Hopcroft, J., and Tarjan, R. E. 1974. Efficient planarity testing. J. ACM 21, 4 (Oct.), 549--568.]] Google Scholar
- Kratochv&ibreave;l, J. 1991. String graphs II: Recognizing string graphs is NP-hard. J. Combinat. Theory Ser. B 52, 1, 67--78.]] Google Scholar
- Ore, O., and Plummer, M. D. 1969. Cyclic coloration of plane graphs. In Recent Progress in Combinatorics (Proceedings of the 3rd Waterloo Conference on Combinatorics, 1968). Academic Press, New York, pp. 287--293.]]Google Scholar
- Renz, J. 1998. A canonical model of the region connection calculus. In Proceedings of the 6th International Conference on Principles of Knowledge Representation and Reasoning. Morgan-Kaufmann, San Francisco, Calif., pp. 330--341.]]Google Scholar
- Schaefer, M., Sedgwick, E., and &Saccute;tefankovi&caccute;, D. 2002. Recognizing string graphs in NP. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC). ACM, New York.]] Google Scholar
- Schaefer, M., and &Saccute;tefankovi&caccute;, D. 2001. Decidability of string graphs. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC). ACM, New York, pp. 241--246.]] Google Scholar
- Thorup, M. 1998. Map graphs in polynomial time. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE, Computer Society Press, Los Alamitos, Calif., pp. 396--407.]] Google Scholar
Index Terms
- Map graphs
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