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Constrained Representations of Map Graphs and Half-Squares

Authors Hoang-Oanh Le, Van Bang Le

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LIPIcs.MFCS.2019.13.pdf
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ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • map graph
  • half-square
  • edge clique cover
  • edge clique partition
  • graph classes

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Abstract

The square of a graph H, denoted H^2, is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B=(X,Y,E_B) are the subgraphs of B^2 induced by the color classes X and Y, B^2[X] and B^2[Y]. For a given graph G=(V,E_G), if G=B^2[V] for some bipartite graph B=(V,W,E_B), then B is a representation of G and W is the set of points in B. If in addition B is planar, then G is also called a map graph and B is a witness of G [Chen, Grigni, Papadimitriou. Map graphs. J. ACM , 49 (2) (2002) 127-138].
While Chen, Grigni, Papadimitriou proved that any map graph G=(V,E_G) has a witness with at most 3|V|-6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown. 
We also consider half-squares of tree-convex bipartite graphs and prove the following complexity dichotomy: Given a graph G=(V,E_G) and an integer k, deciding if G=B^2[V] for some tree-convex bipartite graph B=(V,W,E_B) with |W|<=k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise. Our proof relies on a characterization of half-squares of tree-convex bipartite graphs, saying that these are precisely the chordal and dually chordal graphs.

Cite As Get BibTex

Hoang-Oanh Le and Van Bang Le. Constrained Representations of Map Graphs and Half-Squares. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.13

Author Details

Hoang-Oanh Le
  • Berlin, Germany
Van Bang Le
  • Universität Rostock, Institut für Informatik, Rostock, Germany

References

  1. Franz J. Brandenburg. Characterizing and Recognizing 4-Map Graphs. Algorithmica, 81(5):1818-1843, 2019. URL: https://doi.org/10.1007/s00453-018-0510-x.
  2. Andreas Brandstädt, Feodor F. Dragan, Victor Chepoi, and Vitaly I. Voloshin. Dually Chordal Graphs. SIAM J. Discrete Math., 11(3):437-455, 1998. URL: https://doi.org/10.1137/S0895480193253415.
  3. Maw-Shang Chang and Haiko Müller. On the Tree-Degree of Graphs. In Graph-Theoretic Concepts in Computer Science, 27th International Workshop, WG 2001, Boltenhagen, Germany, June 14-16, 2001, Proceedings, pages 44-54, 2001. URL: https://doi.org/10.1007/3-540-45477-2_6.
  4. Zhi-Zhong Chen. Approximation Algorithms for Independent Sets in Map Graphs. J. Algorithms, 41(1):20-40, 2001. URL: https://doi.org/10.1006/jagm.2001.1178.
  5. Zhi-Zhong Chen, Michelangelo Grigni, and Christos H. Papadimitriou. Planar Map Graphs. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 514-523, 1998. URL: https://doi.org/10.1145/276698.276865.
  6. Zhi-Zhong Chen, Michelangelo Grigni, and Christos H. Papadimitriou. Map graphs. J. ACM, 49(2):127-138, 2002. URL: https://doi.org/10.1145/506147.506148.
  7. Zhi-Zhong Chen, Michelangelo Grigni, and Christos H. Papadimitriou. Recognizing Hole-Free 4-Map Graphs in Cubic Time. Algorithmica, 45(2):227-262, 2006. URL: https://doi.org/10.1007/s00453-005-1184-8.
  8. Zhi-Zhong Chen, Xin He, and Ming-Yang Kao. Nonplanar Topological Inference and Political-Map Graphs. In Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 17-19 January 1999, Baltimore, Maryland, USA., pages 195-204, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.314558.
  9. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs. ACM Trans. Algorithms, 1(1):33-47, 2005. URL: https://doi.org/10.1145/1077464.1077468.
  10. Erik D. Demaine and MohammadTaghi Hajiaghayi. The Bidimensionality Theory and Its Algorithmic Applications. Comput. J., 51(3):292-302, 2008. URL: https://doi.org/10.1093/comjnl/bxm033.
  11. Kord Eickmeyer and Ken-ichi Kawarabayashi. FO model checking on map graphs. In Fundamentals of Computation Theory - 21st International Symposium, FCT 2017, Bordeaux, France, September 11-13, 2017, Proceedings, pages 204-216, 2017. URL: https://doi.org/10.1007/978-3-662-55751-8_17.
  12. Rudolf Fleischer and Xiaotian Wu. Edge Clique Partition of K₄-Free and Planar Graphs. In Computational Geometry, Graphs and Applications - 9th International Conference, CGGA 2010, Dalian, China, November 3-6, 2010, Revised Selected Papers, pages 84-95, 2010. URL: https://doi.org/10.1007/978-3-642-24983-9_9.
  13. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of Map Graphs with Applications. CoRR, abs/1903.01291, 2019. URL: http://arxiv.org/abs/1903.01291.
  14. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1563-1575, 2012. URL: https://doi.org/10.1137/1.9781611973099.124.
  15. Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980. URL: https://doi.org/10.1016/C2013-0-10739-8.
  16. Ian Holyer. The NP-Completeness of Some Edge-Partition Problems. SIAM J. Comput., 10(4):713-717, 1981. URL: https://doi.org/10.1137/0210054.
  17. Lawrence T. Kou, Larry J. Stockmeyer, and C. K. Wong. Covering Edges by Cliques with Regard to Keyword Conflicts and Intersection Graphs. Commun. ACM, 21(2):135-139, 1978. URL: https://doi.org/10.1145/359340.359346.
  18. Hoàng-Oanh Le and Van Bang Le. Hardness and structural results for half-squares of restricted tree-convex bipartite graphs. Algorithmica, in press, 2019. URL: https://doi.org/10.1007/s00453-018-0440-7.
  19. Hoàng-Oanh Le and Van Bang Le. Map graphs having witnesses of large girth. Theor. Comput. Sci., 772:143-148, 2019. URL: https://doi.org/10.1016/j.tcs.2018.12.010.
  20. Van Bang Le and Sheng-Lung Peng. On the complete width and edge clique cover problems. J. Comb. Optim., 36(2):532-548, 2018. URL: https://doi.org/10.1007/s10878-016-0106-9.
  21. Tian Liu. Restricted Bipartite Graphs: Comparison and Hardness Results. In Algorithmic Aspects in Information and Management - 10th International Conference, AAIM 2014, Vancouver, BC, Canada, July 8-11, 2014. Proceedings, pages 241-252, 2014. URL: https://doi.org/10.1007/978-3-319-07956-1_22.
  22. S. Ma, Walter D. Wallis, and Julin Wu. On the complexity of the clique partition problem. Congressus Numerantium, 67:59-66, 1988. Google Scholar
  23. S. Ma, Walter D. Wallis, and Julin Wu. Clique Covering of Chordal Graphs. Utilitas Mathematica, 36:151-152, 1989. Google Scholar
  24. Terry A. McKee and F. R. McMorris. Topics in Intersection Graph Theory. SIAM, 1999. URL: https://epubs.siam.org/doi/book/10.1137/1.9780898719802.
  25. Matthias Mnich, Ignaz Rutter, and Jens M. Schmidt. Linear-time recognition of map graphs with outerplanar witness. Discrete Optimization, 28:63-77, 2018. URL: https://doi.org/10.1016/j.disopt.2017.12.002.
  26. James Orlin. Contentment in graph theory: covering graphs with cliques. Indagationes Mathematicae, 80:406-424, 1977. URL: https://doi.org/10.1016/1385-7258(77)90055-5.
  27. Jeremy P. Spinrad. Efficient Graph Representations. Fields Institute Monographs, 2003. Google Scholar
  28. Mikkel Thorup. Map Graphs in Polynomial Time. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 396-405, 1998. URL: https://doi.org/10.1109/SFCS.1998.743490.
  29. Ryuhei Uehara. NP-complete problems on a 3-connected cubic planar graph and their applications. Tokyo Woman’s Christian University, Technical Report TWCU-M-0004, 1996/9. URL: http://www.jaist.ac.jp/~uehara/pdf/triangle.pdf.
  30. W. D. Wallis and Julin Wu. On clique partitions of split graphs. Discrete Mathematics, 92(1-3):427-429, 1991. URL: https://doi.org/10.1016/0012-365X(91)90297-F.
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