For a graph G on vertex set V = {1, …, n} let
k = (k1, …, kn)
be an integral vector such that
1 [les ] ki [les ] di for
i ∈ V, where di is the degree of the vertex
i in G. A k-dominating set is a set
Dk ⊆ V such that every vertex
i ∈ V[setmn ]Dk has at least
ki neighbours in Dk. The
k-domination number γk(G) of G
is the cardinality of a smallest k-dominating set of G.For k1 = · · · = 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
by N. Alon and J. H. Spencer.If ki = di for
i = 1, …, n, then the notion of k-dominating set corresponds to the
complement of an independent set. A function fk(p)
is defined, and it will be proved that
γk(G) = min fk(p),
where the minimum is taken over the n-dimensional cube
Cn = {p = (p1, …, pn)
[mid ] pi ∈ ℝ, 0 [les ] pi
[les ] 1, i = 1, …, n}. An
[Oscr ](Δ22Δn-algorithm is presented, where Δ
is the maximum degree of G, with INPUT: p ∈ Cn
and OUTPUT: a k-dominating set Dk of G with
[mid ]Dk[mid ][les ]fk(p).