Weak {2}-domination number of Cartesian products of cycles

Abstract

For a graph $G=(V, E)$, a weak $\{2\}$-dominating function $f:V\rightarrow \{0,1,2\}$ has the property that $\sum _{u\in N(v)}f(u)\ge 2$ for every vertex $v\in V$ with $f(v)= 0$, where N(v) is the set of neighbors of v in G. The weight of a weak $\{2\}$-dominating function f is the sum $\sum _{v\in V}f(v)$ and the minimum weight of a weak $\{2\}$-dominating function is the weak $\{2\}$-domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak $\{2\}$-domination number of $C_n \Box C_5$, which was obtained by Stȩpień, et al. (Discrete Appl Math 170:113–116, 2014). Moreover, we obtain the weak $\{2\}$-domination numbers of $C_n \Box C_3$ and $C_n \Box C_4$.