On Grundy total domination number in product graphs

A longest sequence $(v_1,\ldots,v_k)$ of vertices a graph $G$ is Grundy total dominating if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ the called domination number and denoted $\gamma_{gr}^{t}(G)$. In this paper, studied on four standard products. For direct product we show that $\gamma_{gr}^t(G\times H) \geq \gamma_{gr}^t(G)\gamma_{gr}^t(H)$, conjecture equality always holds, prove in several special cases. lexicographic express $\gamma_{gr}^t(G\circ H)$ terms related invariant factors find some explicit formulas it. strong product, lower bounds $\gamma_{gr}^t(G \boxtimes are proved as well upper products paths cycles. Cartesian when or