On total dominating sets in graphs

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if every vertex $v\in V$ is adjacent to an element of $S$. The domination number of a graph $G$ denoted by $\gamma(G)$ is the minimum cardinality of a dominating set in $G$. Respectively the total domination number of a graph $G$ denoted by $\gamma_t(G)$ is the minimum cardinality of a total dominating set in $G$. An upper bound for $\gamma_t(G)$ which has been achieved by Cockayne and et al. in $\cite{coc}$ is: for any graph $G$ with no isolated vertex and maximum degree $\Delta(G)$ and $n$ vertices, $\gamma_t(G)\leq n-\Delta(G)+1$. Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for $\gamma_t(G)$. Also, for circular complete graphs, we determine the value of $\gamma_t(G)$.