Characterizations of the graphs with dominating parameters

A subset $S$ of vertices of $G$ is a \textit{dominating set} of $G$ if every vertex in $V(G)-S$ has a neighbor in $S$. The \textit{domination number} $\gamma(G)$ is the minimum cardinality of a dominating set of $G$. A dominating set $S$ is a \textit{total dominating set} if $N(S)$=$V$ where $N(S)$ is the neighbor of $S$. The \textit{total domination number} $\gamma_t(G)$ equals the minimum cardinality of a total dominating set of $G$. A set $D$ is an \textit{isolate set} if the induced subgragh $G[D]$ has at least one isolated vertex. The \textit{isolate 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.