Characterizations of the graphs with dominating parameters

A subset $S$ of vertices of $G$ is a $\mathit{\text{dominating}}\mathit{\text{set}}$ of $G$ if every vertex in $V(G)-S$ has a neighbor in $S$. The $\mathit{\text{domination}}\mathit{\text{number}}$ $\gamma (G)$ is the minimum cardinality of a dominating set of $G$. A dominating set $S$ is a $\mathit{\text{total}}\mathit{\text{dominating}}\mathit{\text{set}}$ if $N(S)$=$V$ where $N(S)$ is the neighbor of $S$. The $\mathit{\text{total}}\mathit{\text{domination}}\mathit{\text{number}}$ ${\gamma }_t(G)$ equals the minimum cardinality of a total dominating set of $G$. A set $D$ is an $\mathit{\text{isolate}}\mathit{\text{set}}$ if the induced subgragh $G[D]$ has at least one isolated vertex. The $\mathit{\text{isolate}}\mathit{\text{number}}$ $i_0(G)$ is the minimum cardinality of a maximal isolate set. In this paper we study these parameters and answer open problems proposed by Hamid et al. in 2016.