Weak signed Roman domination in graphs

Document Type : Original paper

Author

RWTH Aachen University

Abstract

A weak signed Roman dominating function (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as a function $f:V(G)\rightarrow\{-1,1,2\}$ having the property that $\sum_{x\in N[v]}f(x)\ge 1$ for each $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices. The weak signed Roman domination number of $G$, denoted by $\gamma_{wsR}(G)$, is the minimum weight of a WSRDF in $G$. We initiate the study of the weak signed Roman domination number, and we present different sharp bounds on $\gamma_{wsR}(G)$. In addition, we determine the weak signed Roman domination number of some classes of graphs.

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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] N. Dehgardi and L. Volkmann, Signed total Roman k-domination in directed graphs, Commun. Comb. Optim. 1 (2016), no. 2, 165–178.
[4] N. Dehgardi and L. Volkmann, Nonnegative signed Roman domination in graphs, J. Combin. Math. Combin. Comput. (to appear).
[5] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.
[6] M.A. Henning and L. Volkmann, Signed Roman $k$-domination in trees, Discrete Appl. Math. 186 (2015), 98–105.
[7] M.A. Henning and L. Volkmann, Signed Roman $k$-domination in graphs, Graphs Combin. 32 (2016), no. 1,
175–190.
[8] S.M. Sheikholeslami, R. Khoeilar, and L. Asgharsharghi, Signed strong Roman domination in graphs, Tamkang J. Math. 48 (2017), no. 2, 135–147.
[9] L. Volkmann, On the signed total Roman domination and domatic numbers of graphs, Discrete Appl. Math. 214 (2016), 179–186.
[10] L. Volkmann, Signed total Roman domination in graphs, J. Comb. Optim. 32 (2016), no. 3, 855–871.
[11] L. Volkmann, Signed total Roman k-domination in graphs, J. Combin. Math. Combin. Comput. 105 (2018), 105–116.
 
Communications in Combinatorics and Optimization
Volume 5, Issue 2
December 2020
Pages 111-123

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