On Dominating Sets and Independent Sets of Graphs

  • Authors:

  • Jochen Harant;Anja Pruchnewski;Margit Voigt

  • Affiliations:

  • Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau, Germany (e-mail: harant@mathematik.TU-Ilmenau.de);Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau, Germany (e-mail: harant@mathematik.TU-Ilmenau.de);Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau, Germany (e-mail: harant@mathematik.TU-Ilmenau.de)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 1999

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Abstract

For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 ≤ ki ≤ 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∖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) ∣ pi ∈ ℝ, 0 ≤ pi ≤ 1, i = 1, …, n}. An 𝒪(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 ∣Dk∣≤fk(p).