Upper Bounds on the Total Domination Number

Abstract

A total dominating set of a graph $G$ with no isolated vertex is a set $S$ of vertices of $G$ such that every vertex is adjacent to a vertex in $S$. The total domination number of $G$ is the minimum cardinality of a total dominating set in $G$. In this paper, we present several upper bounds on the total domination number in terms of the minimum degree, diameter, girth, and order.