PressOn Dominating Sets and Independent Setsof Graphs

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the notion of k-dominating set corresponds to the complement of an independent set. A function f k (p) is deened, and it will be proved that k (G) = min f k (p), where the minimum is taken over the n-dimensional cube C(2 2 n)-algorithm is presented, where is the maximum degree of G, with INPUT: p 2 C n and OUTPUT: a k-dominating set D k of G with jD k j f k (p).