The literature on changing and unchanging domination number discusses about the changes in the minimum dominating set on a graph by vertex, edge addition and deletion. Edge contraction and edge subdivision are in some sense duals of each other. In the former case the number of vertices is reduced by one and the number of edges is reduced at least by one while,- in the latter case the number of vertices and edges is increased by one. This property of the two graph operations provide a scope to study and compare when and how the minimum dominating set of a graph does not change, on applying these graph operations. This book is dedicated entirely to introduce and explore the stable property of the minimum dominating set of a graph G with respect to the two graph operations. It also relates them to Excellent Graphs. This book would be of interest to researchers working in domination theory.