Algorithm aspect on total Roman $\{2\}$-domination number of Cartesian products of paths and cycles

A total Roman $\{2\}$-dominating function (TR2DF) on a graph $G$ with vertex set $V$ is a function $f: V\rightarrow \{0,1,2\}$ having the  property that for every vertex $v$ with $f(v)=0$, $\sum_{u\in N(v)}f(u)\geq 2$, where $N(v)$ represents the open neighborhood of $v$, and  the subgraph of $G$ induced by the set of vertices with $f(v)>0$ has no isolated vertex. The weight of a TR2DF $f$ is the value $w(f)=\sum_{v\in V} f(v)$, and the minimum weight of a TR2DF of $G$ is the total Roman $\{2\}$-domination number $\gamma_{tR2}(G)$. The total Roman $\{2\}$-domination problem (TR2DP) is to determine the value $\gamma_{tR2}(G)$. In this paper, we first propose an integer linear programming (ILP) formulation for the TR2DP. Furthermore, we apply the discharging approach to determine the total Roman $\{2\}$-domination number for some Cartesian products of paths and cycles.