| Summary: | A set $D$ of vertices in a graph $G=(V,E)$ is a total dominating<br />set if every vertex of $G$ is adjacent to some vertex in $D$. A<br />total dominating set $D$ of $G$ is said to be weak if every<br />vertex $v\in V-D$ is adjacent to a vertex $u\in D$ such that<br />$d_{G}(v)\geq d_{G}(u)$. The weak total domination number<br />$\gamma_{wt}(G)$ of $G$ is the minimum cardinality of a weak<br />total dominating set of $G$. A total dominating set $D$ of $G$ is<br />said to be strong if every vertex $v\in V-D$ is adjacent to a<br />vertex $u\in D$ such that $d_{G}(v)\leq d_{G}(u)$. The strong<br />total domination number $\gamma_{st}(G)$ of $G$ is the minimum<br />cardinality of a strong total dominating set of $G$. We present<br />some bounds on weak and strong total domination number of a graph.
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