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).