Discussiones Mathematicae Graph Theory 25 (2005 ) 291–302 EXACT DOUBLE DOMINATION IN GRAPHS Mustapha Chellali Department of Mathematics, University of Blida B.P. 270, Blida, Algeria e-mail: mchellali@hotmail.com Abdelkader Khelladi Department of Operations Research Faculty of Mathematics University of Sciences and Technology Houari Boumediene B.P. 32, El Alia, Bab Ezzouar, Algiers, Algeria e-mail: kader khelladi@yahoo.fr and Fr´ ed´ eric Maffray C.N.R.S., Laboratoire Leibniz-IMAG 46 Avenue F´elix Viallet 38031 Grenoble Cedex, France e-mail: frederic.maffray@imag.fr Abstract In a graph a vertex is said to dominate itself and all its neighbours. A doubly dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. A doubly dominating set is exact if every vertex of G is dominated exactly twice. We prove that the existence of an exact doubly dominating set is an NP-complete problem. We show that if an exact double dominating set exists then all such sets have the same size, and we establish bounds on this size. We give a constructive characterization of those trees that admit a doubly dominating set, and we establish a necessary and sufficient condition for the existence of an exact doubly dominating set in a connected cubic graph. Keywords: double domination, exact double domination. 2000 Mathematics Subject Classification: 05C69.