DOMINATION IN THE TOTAL GRAPH ON ℤ_n

Abstract

For a commutative ring R, let Z(R) be its set of zero-divisors. The total graph of R, denoted by T Γ (R), is the undirected graph with vertex set R, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir studied about the domination in the total graph of a commutative ring R. In particular, it was proved that the domination number γ(T Γ (ℤ n )) = p 1 where p 1 is the smallest prime divisor of n. In this paper, we characterize all the γ-sets in T Γ (ℤ n ). Also, we obtain the values of other domination parameters like independent, total and perfect domination numbers of the total graph on ℤ n .