Dominating a Family of Graphs with Small Connected Subgraphs

Yair Caro, Raphael Yuster

Research output: Contribution to journalArticlepeer-review


Let F = {Gl,...,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 υ ∈ V and for each i = 1,...,t, there are at least k neighbours of υ 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=1 δi. Clearly, an (F,k)-core exists if and only if δ(F) ≥ k, and a connected (F,k)-core exists if and only if δ(F) ≥ 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 < √ln δ: c(k, F) ≤ nln δ/δ(1 + oδ(1)), cc(k, F) ≤ nln δ/δ(1 + oδ(1)). 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.

Original languageEnglish
Pages (from-to)309-313
Number of pages5
JournalCombinatorics Probability and Computing
Issue number4
StatePublished - 2000

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics
  • Applied Mathematics


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