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 $γ(G)$ is the minimum cardinality of a dominating set in $G$. Respectively the total domination number of a graph $G$ denoted by $γ_t(G)$ is the minimum cardinality of a total dominating set in $G$. An upper bound for $γ_t(G)$ which has been achieved by Cockayne and et al. in $\cite{coc}$ is: for any graph $G$ with no isolated vertex and maximum degree $Δ(G)$ and $n$ vertices, $γ_t(G)\leq n-Δ(G)+1$. Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for $γ_t(G)$. Also, for circular complete graphs, we determine the value of $γ_t(G)$.