Let Y be a subset of real numbers. A Y dominating function of a graph G = (V, E) is a function f : V → Y such that u∈NG[v] f(u) ≥ 1 for all vertices v ∈ V , where NG[v] = {v} ∪ {u|(u, v) ∈ E}. Let f(S) = u∈S f(u) for any subset S of V and let f(V ) be the weight of f . The Y -domination problem is to find a Y -dominating function of minimum weight for a graph. In this paper, we study the variat...

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 $\gamma_{gr}^t(G\times H) \geq \gamma_{gr}^t(G)\gamma_{gr}^t(H)$, conjec...