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In this paper, we consider families of graphs for which there is a partition of the vertices in a tree of separators associated with a separator theorem. First, we give some séparation properties for the family of quotient graphs with regard to this partition of vertices; then, by adding a bounded degree assumption on the initial family of graphs and using a nested dissection ordering for their vertices, we prove thaï the élimination graph ofsuch a quotient graph is a bounded density graph. Next, we apply these theoretic results to achieve an algorithm of construction of the communication graph associated with a distributed implèmentation of an algorithm of resolution of large sparse Systems of linear équations based on a nested dissection method. This graphy which describes the exchanges between the N processes of the distributed algorithm, is so constructed in O (N) time. (*) Reçu octobre 1986, révisé mai 1987. l) U.E.R. de Mathématiques et d'Informatique, Unité associée au C.N.R.S. n° 226, Université de Bordeaux-I, 351, cours de la Libération, 33405 Talence Cedex. () ENSERB et U.E.R. de Mathématiques et d'Informatique, Unité associée au C.N.R.S. n° 226, Université de Bordeaux-I, 351, cours de la Libération, 33405 Talence Cedex. Informatique théorique et Applications/Theoretical Informaties and Applications 0296-1598 88/02 245 21/84.10/© Gauthier-Villars 246 P. CHARRIER, J. ROMAN

Étude de la séparation et de l'élimination sur une famille de graphes quotients déduite d'une méthode de dissections emboîtées