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Dominating a Family of Graphs with Small Connected Subgraphs

Published online by Cambridge University Press:  03 November 2000

YAIR CARO  and
RAPHAEL YUSTER
YAIR CARO
Affiliation:
Department of Mathematics, University of Haifa-Oranim, Tivon 36006, Israel (e-mail: yairc@macam98.ac.il, raphy@macam98.ac.il)
RAPHAEL YUSTER
Affiliation:
Department of Mathematics, University of Haifa-Oranim, Tivon 36006, Israel (e-mail: yairc@macam98.ac.il, raphy@macam98.ac.il)
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Abstract

Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) [ges ] k, and a connected (F, k)-core exists if and only if δ(F) [ges ] k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:

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The results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 9 , Issue 4 , July 2000 , pp. 309 - 313
Copyright
2000 Cambridge University Press

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