Roman Domination in the Shadow Distance Graphs
DOI:
https://doi.org/10.26713/cma.v14i4.2588Keywords:
Roman dominating function, Roman domination number, Shadow graph, Shadow distance graphAbstract
A function \(\psi: \mathcal{V} \rightarrow \{0,1,2\}\) satisfying the requirement that each vertex \(x\) for which \(\psi(x)=0\) is adjacent to at least one vertex \(y\) for which \(\psi(y)=2\) is known as a Roman dominating function (Rdf) on a graph. A Rdf's weight is represented by the value \(\psi(y)= \sum_{x\in \mathcal{V}} \psi(x)\). The Roman domination number (Rdn) of a graph \(\mathcal{G}\) is the minimal weight of a Rdf on that graph. In this article, we establish Rdn for the shadow distance graph of the path, cycle, and star graphs with predetermined distance sets.
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H. A. Ahangar, A. Bahremandpour, S. M. Sheikholeslami, N. D. Soner, Z. Tahmasbzadehbaee and L. Volkmann, Maximal Roman domination numbers in graphs, Utilitas Mathematica 103 (2017), 245 – 258.
H. A. Ahangar, T. W. Haynes and J. C. Valenzuela-Tripodoro, Mixed Roman domination in graphs, Bulletin of the Malaysian Mathematical Sciences Society 40 (2017), 1443 – 1454, DOI: 10.1007/s40840-015-0141-1.
R. A. Beeler, T. W. Haynes and S. T. Hedetniemi, Double Roman domination, Discrete Applied Mathematics 211 (2016), 23 – 29, DOI: 10.1016/j.dam.2016.03.017.
M. Chellali, T. W. Haynes, S. T. Hedetniemi and A. A. McRae, Roman {2}-domination, Discrete Applied Mathematics 204 (2016), 22 – 28, DOI: 10.1016/j.dam.2015.11.013.
E. J. Cockayne, P. A. Dreyer Jr., S. M. Hedetniemi and S. T. Hedetniemi, Roman domination in graphs, Discrete Mathematics 278(1-3) (2004), 11 – 12, DOI: 10.1016/j.disc.2003.06.004.
M. A. Henning and S. T. Hedetniemi, Defending the Roman empire — A new strategy, Discrete Mathematics 266(1-3) (2003), 239 – 251, DOI: 10.1016/S0012-365X(02)00811-7.
U. V. Kumar and R. Murali, Edge domination in shadow distance graphs, International Journal of Mathematics and its Applications 4(2D) (2016), 125 – 130, URL: http://ijmaa.in/index.php/ijmaa/article/view/1075/1059.
U. V. Kumar and R. Murali, s-Path domination in shadow distance graphs, Journal of Harmonized Research in Applied Sciences 6(3) (2018), 194 – 199.
A. Mekala, U. V. C. Kumar and R. Murali, Bi-domination in brick product graphs, Journal of lgebraic Statistics 13(2) (2022), 1954 – 1960, URL: https://publishoa.com/index.php/journal/article/view/376/347.
C. S. ReVelle and K. E. Rosing, Defendens imperium Romanum: A classical problem in military strategy, The American Mathematical Monthly 107(7) (2000), 585 – 595, DOI: 10.1080/00029890.2000.12005243.
I. Stewart, Defend the Roman empire!, Scientific American 281(6) (1999), 136 – 138, URL: https: //www.jstor.org/stable/26058532.
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