Clique-width of graphs defined by one-vertex extensions

Let GG be a graph and uu be a vertex of GG. We consider the following operation: add a new vertex vv such that vv does not distinguish any two vertices which are not distinguished by uu. We call this operation a one-vertex extension. Adding a true twin, a false twin or a pendant vertex are cases of one-vertex extensions. Here we are interested in graph classes defined by a subset of allowed one-vertex extension. Examples are trees, cographs and distance-hereditary graphs. We give a complete classification of theses classes with respect to their clique-width. We also introduce a new graph parameter called the modular-width, and we give a relation with the clique-width.