Original Research Article

Article volume = 2022 and issue = 1

Pages: 7–10

Article publication Date: November 21, 2022

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Italian domination number upon vertex and edge removal

S. Nazari-Moghaddam

Ilam University, Ilam, Iran.

Abstract:

For a graph G = (V, E), italian domination function f : V -> {0, 1, 2} has the property that for every vertex v &#8712; V with f(v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to at least two vertices assigned 1 under f. The weight of an italian domination function is the sum of its function values over all vertices. The italian domination number &#611;<sub>I</sub>(G) equals the minimum weight of an italian dominating function on G. In this paper, we consider the effects of vertex and edge removal on the Italian domination number of a graph. In addition, we characterize the family of cartesian product of some graphs in terms of Italian domination number.

Keywords:

Edge removal, Italian domination number, Italian dominating function, Graph, Vertex removal.

References:
  • [1] H. Abdollahzadeh Ahangar, T.W. Haynes and J.C. Valenzuela-Tripodoro, Mixed Roman domination in graphs, Bull. Malays. Math. Sci. Soc., 93(7) (2015), 1093–1102. 1
  • [2] H. Abdollahzadeh Ahangar, A. Bahremandpour, S.M. Sheikholeslami, N.D. Soner, Z. Tahmasbzadehbaee and L. Volkmann, Maximal Roman domination numbers in graphs, Util. Math., 103 (2017), 245–258. 1
  • [3] R.A. Beeler, T.W. Haynes and S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math., 211 (2016), 23–29. 1
  • [4] M. Chellali, T. Haynes, S.T. Hedetniemi, A. McRae, Roman {2}-domination, Discrete Appl. Math., 204 (2016), 22–28.
  • [5] E. W. Chambers, B. Kinnersley, N. Prince, and D. B. West Extremal Problems for Roman Domination, SIAM J. Discrete Math., 23 (2009), 1575–1586. 1, 1, 2.7
  • [6] E.J. Cockayne, P.A. Dreyer Jr., S.M. Hedetniemic and S.T. Hedetniemic, Roman domination in graphs, Discrete Math., 278 (2004), 11-22. 1
  • [7] M. Adabi, E. Ebrahimi Targhi, N. Jafari Rad, M. Saied Moradi, Properties of independent Roman domination in graphs, Australas. J. Combin., 52 (2012), 11–18. 1
  • [8] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York (1998). 1
  • [9] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater (eds.), Domination in Graphs: Advanced Topics, Marcel Dekker, New York (1998). 1
  • [10] M.A. Henning and S.T. Hedetniemi, Defending the roman empire–a new strategy, Discrete Math., 266 (2003), 239–251. 1
  • [11] N. Jafari Rad and L. Volkmann, Changing and unchanging the Roman domination number of a graph, Util. Math., 89 (2012), 79–95. 1
  • [12] S. Nazari-Moghaddam and L. Volkmann, Critical concept for double Roman domination in graphs, Discrete Math. Algorithms Appl., 12 (2) (2020), 2050020. 1
  • [13] C.S. Revelle and K.E. Rosing, Defendens imperium romanum: a classical problem in military strategy, Amer. Math. Monthly., 107 (7) (2000), 585–594. 1
  • [14] I. Stewart, Defend the Roman Empire, Sci. Amer., 281 (6) (1999), 136-139. 1
Cite this article as:
  • S. Nazari-Moghaddam, Italian domination number upon vertex and edge removal, Communications in Combinatorics, Cryptography & Computer Science, 2022(1), PP.7–10, 2022
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