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ll graphs considered in our study are simple, finite and undirected. A graph is equitable total domination edge addition critical (stable) if the addition of any arbitrary edge changes (does not change) the equitable total domination number. In this paper, we introduce the following new parameters: equitable independent dom- ination number, equitable total domination number and equitable connected domination number and study their stability upon edge addition, on special families of graphs namely cycles, paths and com- plete bipartite graphs. Also the relation among the above parameters is established.