On total domination and total equitable domination in graphs

Abstract

A dominating set $D$ of a graph $G$ is called total if every vertex of $V(G)$ is adjacent to at least one vertex of $D$, equivalently if $N(D)=V(G)$ then $D$ is called total dominating set. A dominating set $D$ is called total equitable dominating set if it is total and for every vertex in $V(G)-D$ there exists a vertex in $D$ such that they are adjacent and difference between their degrees is at most one. The minimum cardinality of a total (total equitable) dominating set is called total (total equitable) domination number of $G$ which is denoted by $\gamma_t(G)\left(\gamma_t^e(G)\right)$. We have investigated exact value of these parameters for some graphs.