The six classes of graphs resulting from the changing or unchanging of the domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. Each of these classes has been studied individually in the literature. We consider relationships among the classes, which are illustrated in a Venn diagram. We show that no subset of the Venn diagram is empty for arbitrary graphs, and prove that some of the subsets are empty for connected graphs. Our main result is a characterization of trees in each subset of the Venn diagram.

Let uxv be an induced path with center x in a graph G. The edge lifting of uv off x is defined as the action of removing edges ux and vx from the edge set of G, while adding the edge uv to the edge set of G. We study trees for which every possible edge lift changes the domination number. We show that there are no trees for which every possible edge lift decreases the domination number. Trees for which every possible edge lift increases the domination number are characterized.

A graph G is domination bicritical if the removal of any pair of vertices decreases the domination number. Properties of bicritical graphs are studied. We show that a connected bicritical graph has domination number at least 3, minimum degree at least 3, and edge-connectivity at least 2. Ways of constructing a bicritical graph from smaller bicritical graphs are presented.