Bounds on weak and strong total domination in graphs

Abstract

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