Let G (V, E) be a finite undirected graph with n vertices and m edges. A minimum edge dominating set of G is a set of edges D, of smallest cardinality 3"(G), such that each edge of E D is adjacent to someThe edge domination problem is NP-complete for planar bipartite graphs, their subdivision, line, and total graphs, perfect claw-free graphs, and planar cubic graphs. The stable set problem and the edge domination problem are NP-complete for iterated total graphs.The edge domination problem is solvable in O(n 3) time for claw-free chordal graphs, locally connected claw-free graphs, the line graphs of total graphs, the line graphs of chordal graphs, the line graph of any graph in which each nonbridge edge is in a triangle, and the total graphs of any of the preceding graphs.

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The problem of adaptive multicolouring is finding a multicolouring for a graph and each of a sequence of changing weight vectors. Recolouring is either not allowed at all, or is allowed only in limited amounts. The aim is to minimize the number of colours used, subject to certain restrictions on the weight vectors. We establish the number of colours needed for adaptive multicolouring for the class of k-colourable graphs, both for the case when no recolouring is allowed, and for a case where limited recolouring is allowed. Subject Classification (1991): 05C15 * This work was partially financed by the FCAR-BNR-CRSNG project "Méthodes mathématiques pour la synthèse des systèmes informatiques." Mathematics