Domination edge lift critical trees

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.