Weak signed Roman $k$-domination in graphs

Document Type : Original paper


RWTH Aachen University


Let $k\ge 1$ 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 $f:V(G)\rightarrow\{-1,1,2\}$ satisfying the conditions that $\sum_{x\in N[v]}f(x)\ge k$ for each vertex $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is $w(f)=\sum_{v\in V(G)}f(v)$. The weak signed Roman $k$-domination number $\gamma_{wsR}^k(G)$ 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 $\gamma_{wsR}^k(G)$. 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 $\gamma_{sR}^k(G)$, introduced and investigated by Henning and Volkmann [5] as well as Ahangar, Henning, Zhao, Löwenstein and Samodivkin [1] for the case $k=1$.


Main Subjects

[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.