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Realizability of Graphs as Triangle Cover Contact Graphs

  • Shaheena Sultana
  • Md. Saidur Rahman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10043)

Abstract

Let \(S=\{p_1,p_2,\ldots ,p_n\}\) be a set of pairwise disjoint geometric objects of some type and let \(C=\{c_1,c_2,\ldots ,c_n\}\) be a set of closed objects of some type with the property that each element in C covers exactly one element in S and any two elements in C can intersect only on their boundaries. We call an element in S a seed and an element in C a cover. A cover contact graph (CCG) consists of a set of vertices and a set of edges where each of the vertex corresponds to each of the covers and each edge corresponds to a connection between two covers if and only if they touch at their boundaries. A triangle cover contact graph (TCCG) is a cover contact graph whose cover elements are triangles. In this paper, we show that every Halin graph has a realization as a TCCG on a given set of collinear seeds. We introduce a new class of graphs which we call super-Halin graphs. We also show that the classes super-Halin graphs, cubic planar Hamiltonian graphs and \(a\times b\) grid graphs have realizations as TCCGs on collinear seeds. We also show that every complete graph has a realization as a TCCG on any given set of seeds. Note that only trees and cycles are known to be realizable as CCGs and outerplanar graphs are known to be realizable as TCCGs.

Notes

Acknowledgement

This work is done in Graph Drawing and Information Visualization Laboratory, Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology as a part of a Ph.D. research work. The first author is supported by ICT Fellowship of ICT division, Ministry of Posts, Telecommunications and IT, Government of the People’s Republic of Bangladesh.

References

  1. 1.
    Abellanas, M., Bereg, S., Hurtado, F., Olaverri, A.G., Rappaport, D., Tejel, J.: Moving coins. Comput. Geom. 34(1), 35–48 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abellanas, M., Castro, N., Hernández, G., Márquez, A., Moreno-Jiménez, C.: Gear System Graphs. Manuscript (2006)Google Scholar
  3. 3.
    Atienza, N., Castro, N., Cortés, C., Garrido, M.A., Grima, C.I., Hernández, G., Márquez, A., Moreno, A., Nöllenburg, M., Portillo, J.R., Reyes, P., Valenzuela, J., Trinidad Villar, M., Wolff, A.: Cover contact graphs. J. Comput. Geom. 3(1), 102–131 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    De Fraysseix, H., de Mendez, P.O.: Representations by contact and intersection of segments. J. Algorithmica 47(4), 453–463 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall Inc., Upper Saddle River (1999)zbMATHGoogle Scholar
  6. 6.
    Durocher, S., Mehrabi, S., Skala, M., Wahid, M.A.: The cover contact graph of discs touching a line. In: CCCG, pp. 59–64 (2012)Google Scholar
  7. 7.
    Hossain, M.I., Sultana, S., Moon, N.N., Hashem, T., Rahman, M.S.: On triangle cover contact graphs. In: Rahman, M.S., Tomita, E. (eds.) WALCOM 2015. LNCS, vol. 8973, pp. 323–328. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-15612-5_29 CrossRefGoogle Scholar
  8. 8.
    Koebe, P.: Kontaktprobleme der konformen abbildung. Ber. Sächs. Akad. Wiss. Leipzig Math.-Phys. Klasse 88(1–3), 141–164 (1936)zbMATHGoogle Scholar
  9. 9.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. Lecture Notes Series on Computing. World Scientific, Singapore (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley, New York (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Robert, J.M., Toussaint, G.T. (eds.): Computational geometry and facility location. In: Proceedings of the International Conference on Operations Research and Management Science, vol. 68 (1990)Google Scholar
  12. 12.
    Welzl, E.: Smallest enclosing disks (balls and ellipsoids). In: Maurer, I.H. (ed.) New Results and New Trends in Computer Science. LNCS, vol. 555, pp. 359–370. Springer, Heidelberg (1991)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Graph Drawing and Information Visualization Laboratory, Department of Computer Science and EngineeringBangladesh University of Engineering and Technology (BUET)DhakaBangladesh

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