Bounds on weak and strong total domination in graphs

M.H. Akhbari (Department of Mathematics, Estahban Branch, Islamic Azad University, Estahban, Iran)
Nader Jafari Rad (Department of Mathematics, Shahrood University of Technology, Shahrood, Iran)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 4, No 1 (2016): Electronic Journal of Graph Theory and Applications

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$d_{G}(v)\geq d_{G}(u)$. The weak total domination number$\gamma_{wt}(G)$ 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 $d_{G}(v)\leq d_{G}(u)$. The strongtotal domination number $\gamma_{st}(G)$ 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.