Bounds on weak and strong total domination in graphs - M.H. Akhbari; Nader Jafari Rad (Electronic Journal of Graph Theory and Applications (EJGTA), 2016)
A set $D$ of vertices in a graph $G=(V,E)$ is a total dominatingset if every vertex of $G$ is adjacent to some vertex in $D$. Atotal dominating set $D$ of $G$ is said to be weak if everyvertex $v\in V-D$ is adjacent to a vertex $u\in D$ such that$d_{G}(v)≥ d_{G}(u)$. The weak total domination number$γ_{wt}(G)$ of $G$ is the minimum cardinality of a weaktotal dominating set of $G$. A total dominating set $D$ of $G$ issaid to be strong if every vertex $v\in V-D$ is adjacent to avertex $u\in D$ such that $d_{G}(v)≤ d_{G}(u)$. The strongtotal domination number $γ_{st}(G)$ of $G$ is the minimumcardinality of a strong total dominating set of $G$. We presentsome bounds on weak and strong total domination number of a graph.