The Numerical Invariants concerning the Total Domination for Generalized Petersen Graphs

Abstract

A subset $S$ of $V\left(G\right)$ is called a total dominating set of a graph $G$ if every vertex in $V\left(G\right)$ is adjacent to a vertex in $S$ . The total domination number of a graph $G$ denoted by ${\gamma }_t\left(G\right)$ is the minimum cardinality of a total dominating set in $G$ . The maximum order of a partition of $V\left(G\right)$ into total dominating sets of $G$ is called the total domatic number of $G$ and is denoted by $d_t\left(G\right)$ . Domination in graphs has applications to several fields. Domination arises in facility location problems, where the number of facilities (e.g., hospitals and fire stations) is fixed, and one attempts to minimize the distance that a person needs to travel to get to the closest facility. In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs.