Abstract
In this paper, we investigate the domination number, independent domination number, connected domination number, total domination number denoted by \( \gamma (G\left( n \right)), \gamma_{i} (G\left( n \right)), \gamma_{c} (G\left( n \right),\gamma_{t} (G\left( n \right)) \) respectively for 4-regular graphs of n vertices with girth 3. Here, G(n) denotes the 4-regular graphs of n vertices with girth 3. We obtain some exact values of G(n) for these parameters. We further establish that \( \gamma_{i} \left( {G\left( n \right)} \right) = \gamma \left( {G\left( n \right)} \right)\, {\text{for }} n \ge 6 \) and \( \gamma_{c} \left(G\left( n \right)\right) = \gamma_{t} \left( {G\left( n \right)} \right) \) for n ≥ 6. Nordhaus–Gaddum type results are also obtained for these parameters.
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References
Duckworth W, Mans B (2009) Connected domination of regular graphs. Discrete Math 309:2305–2322
Ramya N, Rangarajan K, Sattanathan R (2011) On Coloring of Bi-Magic 4-Regular Graphs. Journal of Ultra Scientist of Physical Science 23(3)
Thirusangu K, Bala E (2011) On Bi-Magic labeling of 4-regular graphs. Indian J Sci and Technol 4:414–416
Bondy J A, Murty U S R (1976) Graph Theory with Applications, Macmillan Press
Ore O (1962) Theory of Graphs, American Mathematical Society. Colloq. Publ. (American Mathematical Soc. Providence. RI)38
Haynes T W, Hedetniemi S T, Slater P J (1998) Fundamentals of domination in graphs. Marcel Dekker Inc. New York. Vol. 208
Haynes TW, Hedetniemi ST, Slater PJ (1998) Domination in graphs: advanced topics. Marcel Dekker, New York
Cockayne EJ, Hedetniemi ST (1990) Total domination in graphs. Networks 10:211–219
Harary F, Haynes TW (1996) Nordhaus-Gaddum inequalities for domination in graphs. Discrete Math 155:99–105
Paulraj J, Arumugam S (1995) Domination in graphs. Int J Manage Sys 11:177–182
Nordhaus EA, Gaddum JW (1956) On complementary graphs. Am Math Monthly 63:175–177
Jaeger F, Payan C (1972) Relations du type Norhaus-Gaddum pour le nombre d’absorption d’un graph simple. C R Acad Sci Paris 274:728–730
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Mohanapriya, N., Vimal Kumar, S., Vernold Vivin, J. et al. Domination in 4-Regular Graphs with Girth 3. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 85, 259–264 (2015). https://doi.org/10.1007/s40010-015-0201-9
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DOI: https://doi.org/10.1007/s40010-015-0201-9