Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

NORDHAUS-GADDUM TYPE RESULTS FOR CONNECTED DOMINATION NUMBER OF GRAPHS

  • E. Murugan (Department of Mathematics, C.S.I Jayaraj Annapackiam College) ;
  • J. Paulraj Joseph (Department of Mathematics, Manonmaniam Sundaranar University)
  • Received : 2022.07.06
  • Accepted : 2023.10.07
  • Published : 2023.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let G = (V, E) be a graph. A subset S of V is called a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number γ(G) of G is the minimum cardinality taken over all dominating sets of G. A dominating set S is called a connected dominating set if the subgraph induced by S is connected. The minimum cardinality taken over all connected dominating sets of G is called the connected domination number of G, and is denoted by γc(G). In this paper, we investigate the Nordhaus-Gaddum type results for the connected domination number and its derived graphs like line graph, subdivision graph, power graph, block graph and total graph, and characterize the extremal graphs.

Keywords

Acknowledgement

We thank the reviewer for his/her thorough review and appreciate comments and suggestions, which significantly contributed to improving the quality of the publication.

References

  1. M. Aouchiche and P. Hansen, A survey of Nordhaus-Gaddum type relations, Discrete Applied Mathematics. 161 (4-5) (2013), 466-546. https://doi.org/10.1016/j.dam.2011.12.018
  2. S. Arumugam and J. Paulraj Joseph, Domination in Subdivision Graphs, Journal of Indian Math Soc. 62 (1-4) (1996), 274-282.
  3. S. Arumugam and S. Velammal, Connected Edge Domination in Graphs, Bulletin of the Allahabad Mathematical Society Pramila Srivastava Memorial Volume. 24 (1) (2009), 43-49.
  4. M. Behzad, A criterion for the planarity of a total graph, Proc. Cambridge Philos. Soc. 63 (1967), 679-681. https://doi.org/10.1017/S0305004100041657
  5. C. Berge, Theory of Graphs and Its Applications, London, Hethuen, 1962.
  6. J. A. Bondy and U. S. R. Murty, Graph Theory, London, Spinger, 2008.
  7. M. B. Cozzens and L. L. Kelleher, Dominating cliques in graphs, Discrete Math. 86 (1990), 145-164. https://doi.org/10.1016/0012-365X(90)90357-N
  8. Frank Harary, Graph Theory, Addison-Wesley Publishing Company, London, 1969.
  9. Gary Chartrand and Ping Zhang, Introduction To Graph Theory, Tata McGraw-Hill, 2006.
  10. T. W. Haynes, S. T. Hedetnimi and P. J. Slater, Fundamentals of domination in graphs, New York, Marcel Dekkar, Inc., 1998.
  11. S. T. Hedetniemi and R. C. Laskar, Connected domination in graphs, In B. Bollobas, editor, Graph Theory and Combinatorics, Academic Press. London, 1984, 209-218.
  12. F. Jaeger and C. Payan, Relations due Type Nordhaus-Gaddum pour le Nombre d'Absorption d'un Graphe Simple, C. R. Acad. Sci. Paris Ser. A. 274 (1972), 728-730.
  13. B. S. Karunagaram and J. Paulraj Joseph, On Domination Parameters and Maximum Degree of a Graph, Journal of Discrete Mathematical Sciences & Cryptography. 9 (2) (2006), 215-223. https://doi.org/10.1080/09720529.2006.10698073
  14. G. Mahadevan, Selvam Avadayappan, J. Paulraj Joseph, B. Ayisha and T. Subramanian, Complementary Triple Connected Domination Number of a Graph, Advances and Applications in Discrete Mathematics. 12 (1) (2013), 39-54.
  15. Moo Young Sohn, Sang Bum Kim, Young Soo Kwon and Rong Quan Feng, Classification of Regular Planar Graphs with Diameter Two, Acta Mathematica Sinica, English Series. 23 (3) (2007), 411-416. https://doi.org/10.1007/s10114-005-0607-4
  16. E. Murugan and J. Paulraj Joseph, On the domination number of a graph and its line graph, International Journal of Mathematical Combinatorics. Special Issue 1, (2018), 170-181.
  17. E. Murugan and J. Paulraj Joseph, On the Domination Number of a Graph and its Total Graph, Discrete Mathematics, Algorithms and Applications. 12(5) (2020) 2050068.
  18. E. Murugan and G. R. Sivaprakash, On the Domination Number of a Graph and its Shadow Graph, Discrete Mathematics, Algorithms and Applications. 13 (6) (2021) 2150074.
  19. E. Murugan and J. Paulraj Joseph, On the Domination Number of a Graph and its Block Graph, Discrete Mathematics, Algorithms and Applications. 14 (7) (2022) 2250033.
  20. E. A. Nordhaus and J. Gaddum, On complementary graphs, Amer. Math. Monthly. 63 (1956) 177-182. https://doi.org/10.2307/2306658
  21. O. Ore, Theory of Graphs, Am. Math. SOC. Colloq. Publ. 38, Providence, RI, 1962.
  22. E. Sampathkumar and H. B. Walikar, The connected domination number of a graph, J. Math. Phys. Sci. 13 (1979) 607-613.
  23. W. J. Desormeaux, T. W. Haynes and M. A. Henning, Bounds on the connected domination number of a graph, Discrete Applied Mathematics 161 (2013), 2925-2931. https://doi.org/10.1016/j.dam.2013.06.023