Signed total Italian k-domination in graphs

Document Type : Original paper

Author

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 signed total Italian k-ominating function (STIkDF) 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 neighborhood of v, and each vertex u with f(u)=−1 is adjacent to a vertex v with f(v)=2 or to two vertices w and z with f(w)=f(z)=1. The weight of an STIkDF f is ω(f)=∑v∈V(G)f(v). The signed total Italian k-domination number γstIk(G) of G is the minimum weight of an STIkDF on G. In this paper we initiate the study of the signed total Italian k-domination number of graphs, and we  present different bounds on γstIk(G). In addition, we determine the
signed total Italian k-domination number of some classes of graphs.  Some of our results are extensions of well-known properties of the signed total Roman k-domination number γstRk(G), introduced and investigated by Volkmann [9,12].

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References

[1] J. Amjadi and M. Soroudi, Twin signed total Roman domination numbers in digraphs, Asian-European J. Math. 11 (2018), no. 3, 1850034.
[2] E.J. Cockayne, Dawes R.M., and Hedetniemi S.T., Total domination in graphs, Networks 10 (1980), no. 3, 211–219.
[3] N. Dehgardi and L. Volkmann, Signed total Roman k-domination in directed graphs, Commun. Comb. Optim. 1 (2016), no. 2, 165–178.
[4] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc., New York, 1998.
[5] V.R. Kulli, On n-total domination number in graphs, Graph theory, Combinatorics, Algorithms, and Applications (San Francisco, CA, 1989),(SIAM, Philadelphia, PA, 1991) (1991), 319–324.
[6] L. Volkmann, Signed total Italian domination in graphs, J. Combin. Math. Combin. Comput., (to appear).
[7] , On the signed total Roman domination and domatic numbers of graphs, Discrete Appl. Math. 214 (2016), 179–186.
[8] L. Volkmann, Signed total Roman domination in digraphs, Discuss. Math. Graph Theory 37 (2017), no. 1, 261–272.
[9] L. Volkmann, Signed total Roman k-domination in graphs, J. Combin. Math. Combin. Comput. 105 (2018), 105–116.
[10] L. Volkmann, Weak signed Roman domination in graphs, Commun. Comb. Optim. 5 (2020), no. 2, 111–123.
[11] L. Volkmann, Weak signed Roman k-domination in graphs, Commun. Comb. Optim. 6 (2021), no. 1, 1–15.
[12] L. Volkmann, Signed total Roman domination in graphs, J. Comb. Optim. 32 (2016), no. 3, 855–871.
Communications in Combinatorics and Optimization
Volume 6, Issue 2
December 2021
Pages 171-183

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