Central graphs and Italian domination parameters
DOI:
https://doi.org/10.31642/JoKMC/2018/110104%20Keywords:
Italian domination, restrained (total restrained) Italian domination, Central graphAbstract
Abstract Let be a graph with and A function is said to be an Italian dominating function on a graph if every vertex with is adjacent to at least one vertex with or is adjacent to at least two vertices with . The value denotes the weight of an Italian dominating function. The minimum weight taken over all Italian dominating functions of is called Italian domination number and denoted by
Two parameters related to Italian dominating function are restrained Italian and total restrained dominating functions , for which the set of vertices with , and simultaneously the set of vertices with and the set of vertices with induce subgraphs with no isolated vertex respectively. The central graph of a graph is the graph obtained by subdividing each edge of exactly once and joining all the non-adjacent vertices of
In this work, we initiate the study of restrained (total restrained) Italian domination number of the central of any graph For a family of standard graphs we obtain the precise value of restrained (total restrained) Italian domination number for indeed for any graph G, the sharp bounds are provided for and for corona of , we establish the precise value of these parameters for
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M. Chellali, T.W. Haynes, S.T. Hedetniemi and A.A. McRae, Roman {2}-domination, Dis- crete Appl. Math. 204 (2016) 22–28.
E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11–22.
G. S. Domke, J. H. Hattingh, S. T. Hedetniemi, R. C. LaskarcLis, R. Markus, Restrained domination in graphs, Discrete Mathematics, 203(1-3) (1999) 61-69.
M. Furuya, Bounds on the Domination Number of a Digraph and its Reverse, Filomat, 32(7) (2018) 2517-2524.
A. Hansberg, L. Volkmann, B. Randerath, Claw-Free Graphs with Equal 2-Domination and Domination Numbers, Filomat, 30(10) (2016) 2795-2801.
T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, (1998).
M.A. Henning and W.F. Klostermeyer, Italian domination in trees. Discrete Appl. Math. 217.(2017) 557-564.
F. Ramezani, E. Rodriguez-Bazan, J.A. Rodrguez-Velzquez, On the Roman Domination Num- ber of Generalized Sierpinski Graphs, Filomat, 31(20) (2017) 6515-6528.
A.M. Ridha Abdulhasan and D.A. Mojdeh, Further results on (total) restrained italian dom- ination, Discrete Math. Algorithms Appl., (2023) 2350017, DOI: 10.1142/S1793830923500179.
P. Roushini Leely Pushpam and S. Padmapriea, Restrained Roman domination in graphs, Transactions on Combinatorics, 4(1) (2015) 1-17.
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.
I. Stewart, Defend the Roman Empire!, Sci. Amer. 281 (1999) 136–139.
L. Volkmann, Remarks on the restrained Italian domination number in graphs, Communications in Combinatorics and Optimization, 8(1) (2023) 183-191, DOI: 10.22049/CCO.2021.27471.1269.
C. Wang, Domination and Korenblum constants for some function spaces, Filomat, 37(16) (2023).
D.B. West, Introduction to Graph Theory. Second Edition, Prentice-Hall, Upper Saddle River, NJ (2001).
T. Zec, Roman domination problem on Johnson graphs, Filomat, 37(7) (2023) 2067-2075.
.
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