On total domination and total equitable domination in graphs

On total domination and total equitable domination in graphs

**Authors : **

S. K. Vaidya ^{1 *} and A. D. Parmar^{ 2}

**Author Address : **

^{1}Department of Mathematics, Saurashtra University, Rajkot - 360 005, Gujarat, India.

^{2}Atmiya Institute of Technology and Science for Diploma Studies, Rajkot - 360 005, Gujarat, India.

*Corresponding author.

**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 $gamma _t (G)$($gamma _t^e(G)$). We have investigated exact value of these parameters for some graphs.

**Keywords : **

Dominating set, total dominating set, equitable dominating set.

**DOI : **

**Article Info : **

*Received : * November 12, 2017; *Accepted : * March 23, 2018.