Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the numbe...

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...