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Let \( R \) be a commutative ring and \( Z(R) \) be its set of all zero-divisors. The \emph{total graph} of \( R \), denoted by \( T_\Gamma(R) \), is the undirected graph with vertex set \( R \), where two distinct vertices \( x \) and \( y \) are adjacent if and only if \( x + y \in Z(R) \).
In this paper, we obtain a lower bound as well as an upper bound for the domination number of \( T_\Gamma(R) \). Further, we prove that the upper bound for the domination number of \( T_\Gamma(R) \) is attained in the case of an Artin ring \( R \). Having established this, we identify certain classes of rings for which the domination number of the total graph equals this upper bound.
In view of these results, we conjecture that the domination number of \( T_\Gamma(R) \) is always equal to this upper bound. We also derive certain other domination parameters for \( T_\Gamma(R) \) under the assumption that the conjecture is true.