Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-17T17:48:42.237Z Has data issue: false hasContentIssue false

On Dominating Sets and Independent Sets of Graphs

Published online by Cambridge University Press:  01 November 1999

JOCHEN HARANT
Affiliation:
Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau, Germany (e-mail: harant@mathematik.TU-Ilmenau.de)
ANJA PRUCHNEWSKI
Affiliation:
Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau, Germany (e-mail: harant@mathematik.TU-Ilmenau.de)
MARGIT VOIGT
Affiliation:
Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau, Germany (e-mail: harant@mathematik.TU-Ilmenau.de)

Abstract

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 iV, where di is the degree of the vertex i in G. A k-dominating set is a set DkV such that every vertex iV[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: pCn and OUTPUT: a k-dominating set Dk of G with [mid ]Dk[mid ][les ]fk(p).

Type
Research Article
Copyright
© 1999 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)