On the total restrained double Italian domination

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

Abstract

A double Italian  dominating (DID) function  of a graph $G=(V,E)$ is a function $f: V(G)\to\{0,1,2,3\}$ having
the property that for every vertex $v\in V$, $\sum_{u\in N_G[v]}f(u)\geq 3$, if $f(v)\in \{0,1\}$.
A restrained  double Italian dominating (RDID) function is a DID function $f$  such that the subgraph induced by the vertices
with label $0$ has no isolated vertex.
A total restrained double Italian dominating (TRDID) function is an RDID function $f$  such that the set $\{v\in V: f(v)> 0\}$  induces a subgraph with no isolated vertex.\\
We initiate the study of TRDID function of any graph $G$. The TRDID and RDID functions of the middle of any graph $G$ are investigated,
and then,  the sharp bounds for these parameters are established.
Finally, for  a  graph $H$, we provide the minimum value of TRDID and RDID functions for corona graphs,
$H \circ K_1$, $H \circ K_2$ and middle of them.
 

Keywords

Full Text

‎

References

1. F. Azvin and N. Jafari Rad, Bounds on the double italian domination number of a graph, to appear in Discuss. Math. Graph Theory, doi:10.7151/dmgt.2330.
2. F. Azvin, N. Jafari Rad and L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. (6) (1 (2021), 123-136, DOI: 10.22049/CCO.2020.26928.1166.
3. R.A. Beeler, T.W. Haynes and S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016), 23-29, http://dx.doi.org/10.1016/j.dam.2016.03.017.
4. M. Chellali, T.W. Haynes, S.T. Hedetniemi and A.A. McRaee, Roman {2}-domination, Discrete Appl. Math. 204 (2016), 22-28, http://dx.doi.org/10.1016/j.dam.2015.11.013.
5. M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Varieties of Roman domination II, AKCE Int. J. Graphs Comb. (17) 3 (2020), 966-984.
6. E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004), 11-22, http://dx.doi.org/10.1016/j.disc.2003.06.004.
7. G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar, and L.R. Markus, Restrained domination in graphs, Discrete Math. 203 (1-3) (1999), 61-69.
8. G. Hao, L. Volkmann, and D.A. Mojdeh, Total double Roman domination in graphs, Commun. Comb. Optim. (5) 1 (2020), 27-39.
9. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel. Dekker, New York, 1998.
10. J.H. Hattingh and E.F. Joubert, Restrained and total restrained domination in graphs, Topics in Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer, (2020), 129-150.
11. M. A. Henning and W. F. Klostermeyer, Italian domination in trees, Discrete Appl. Math., 217 (2017), 557-564,
http://dx.doi.org/10.1016/j.dam.2016.09.035.
12. R. Jalaei and D.A. Mojdeh, Outer independent double Italian domination: complexity, characterization, Discrete Math., Algorithms and Appl., (15) 3 (2022), 2250085 (13 pages) DOI: 10.1142/S1793830922500859.
13. R. Jalaei and D.A. Mojdeh, Outer independent double Italian domination of some graph products, Theory Appl. Graphs, (10) 1 (2023), Article (5).
14. D.A. Mojdeh and Zh. Mansouri, On the Independent Double Roman Domination in Graphs, Bull. Iran. Math. Soc. 46 (2020), 905-915,https://doi.org/10.1007/s41980-019-00300-9.
15. D.A. Mojdeh, I. Masoumi and L. Volkmann, Restrained double Roman domination of a graph, RAIRO-Oper. Res. 56 (2022), 2293-2304, https://doi.org/10.1051/ro/2022089.
16. D.A. Mojdeh and A. Teymourzadeh, Upper bounds for covering total double Roman domination, accepted by TWMS J. App. and Eng. Math. (2021).
17. D.A. Mojdeh and L. Volkmann, Roman f3g-domination (double Italian domination), Discrete Appl. Math. 283 (2020), 555-564.
18. C.S. ReVelle and K.E. Rosing, Defendens imperium Romanum: a classical problem in military strategy, Amer. Math. Monthly (107) 7 (2000), 585-594, http://dx.doi.org/10.2307/2589113.
19. I. Stewart, Defend the Roman empire!, Sci. Amer. (281) 6 (1999), 136-139.
20. B. Samadi, M. Alishahi, I. Masoumi and D. A. Mojdeh, Restrained italian domination in graphs, RAIRO-Oper. Res. 55 (2021), 319-332, https://doi.org/10.1051/ro/2021022.
21. A. Teymourzadeh and D.A. Mojdeh, Covering total double Roman domination in graphs, CCommun. Comb. Optim. (8) 1 (2023), 115-125, DOI:10.22049/CCO.2021.27443.1265.
22. E. Vatandoost and K. Mirasheh, Three bounds for identifying code number, J. Algebra Relat. Topics, (10) 2 (2022), 61-67, DOI:10.22124/jart.2022.20661.1321.
23. L. Volkmann, Restrained double Italian domination in graphs, Commun. Comb.Optim. (8) 1 (2023), 1-11, Doi:10.22049/CCO.2021.27334.1236.
24. L. Volkmann, Remarks on the restrained Italian domination number in graphs, Commun. Comb. Optim. (8) 1 (2023) 183-191, DOI:10.22049/CCO.2021.27471.1269.
25. D.B. West, Introduction to Graph Theory. Second Edition, Prentice-Hall, Upper Saddle River, NJ 2001.
Journal of Algebra and Related Topics
Volume 12, Issue 1
July 2024
Pages 105-126

Files

Share

How to cite

Statistics