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We consider a discrete-time dynamic problem in graphs in which the goal is to maintain a dominating set over an infinite sequence of time steps. At each time step, a specified vertex in the current dominating set must be replaced by a neighbor. In one version of the problem, the only change to the current dominating set is replacement of the specified vertex. In another version of the problem, other vertices in the dominating set can also be replaced by neighbors. A variety of results are presented relating these new parameters to the eternal domination number, domination number, and independence number of a graph.