γγ-Excellent, critically dominated, end-dominated, and dot-critical trees are equivalent

A graph is γγ-excellent if every vertex of the graph is contained in some minimum dominating set of the graph. A vertex vv is critical in G   if the domination number of G-vG-v is smaller than that of G. The graph G is dot-critical if contracting any edge of G produces a graph with smaller domination number. G is critically dominated if the set of critical vertices forms a dominating set for G. In this paper we show that these three properties, along with several others, are equivalent for trees on at least four vertices. We also provide a constructive characterization of these trees.