Cototal Domination Number of a Zero Divisor Graph

Abstract

Let R be a commutative ring and let Z(R) be its set of zero-divisors. We associate a graph $\Gamma(R)$ to R with vertices $Z(R)^{*}$ = $Z(R)-\{0\}$, the set of non-zero zero divisors of R and for distinct $u,v \in Z(R)^{*}$, the vertices $u$ and $v$ are adjacent if and only if $uv=0$. A dominating set D of G is a total dominating set if the induced subgraph of $\langle D \rangle$ contains no isolated vertices. The total domination number $\gamma_{t}(G)$ of G is the minimum cardinality of a total dominating set. A dominating set D of G is a cototal dominating set if every vertex $v \in V-D$ is not an isolated vertex in $\langle V-D \rangle$. The cototal domination number $\gamma_{ct}(G)$ of G is the minimum cardinality of a cototal dominating set. In this paper, we evaluate the cototal domination number of $\Gamma(Z_{n})$.