Weak signed Roman k-domination in graphs

Document Type : Original paper

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RWTH Aachen University

Abstract

Let k≥1 be an integer, and let G be a finite and simple graph with vertex set V(G). A weak signed Roman k-dominating function (WSRkDF) on a graph G is a function f:V(G)→{−1,1,2} satisfying the conditions that ∑x∈N[v]f(x)≥k for each vertex v∈V(G), where N[v] is the closed neighborhood of v. The weight of a WSRkDF f is w(f)=∑v∈V(G)f(v). The weak signed Roman k-domination number γwsRk(G) of G is the minimum weight of a WSRkDF on G. In this paper we initiate the study of the weak signed Roman k-domination number of graphs, and we present different bounds on γwsRk(G). In addition, we determine the weak signed Roman k-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed Roman k-domination number γsRk(G), introduced and investigated by Henning and Volkmann [5] as well as Ahangar, Henning, Zhao, Löwenstein and Samodivkin [1] for the case k=1.

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References

[1] H. Abdollahzadeh Ahangar, M.A. Henning, C. Löwenstein, Y. Zhao, and V. Samodivkin, Signed Roman domination in graphs, J. Comb. Optim. 27 (2014), no. 2, 241–255.
[2] J. Amjadi, S. Nazari-Moghaddam, S.M. Sheikholeslami, and L. Volkmann, On the signed Roman k-domination in graphs, Quaest. Math. (to appear).
[3] J.F. Fink and M.S. Jacobson, n-domination in graphs, Graph Theory with Applications to Algorithms and Computer Science, John Wiley & Sons, Inc., 1985, pp. 283–300. 
[4] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc., New York, 1998.
[5] M.A. Henning and L. Volkmann, Signed Roman k-domination in graphs, Graphs Combin. 32 (2016), no. 1, 175–190.
[6] S.M. Sheikholeslami and L. Volkmann, Signed Roman domination in digraphs, J. Comb. Optim. 30 (2015), no. 3, 456–467.
[7] L. Volkmann, Weak signed Roman domination in graphs, Commun. Comb. Optim. 5 (2020), no. 2, 111–123.
Communications in Combinatorics and Optimization
Volume 6, Issue 1
June 2021
Pages 1-15

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