## Abstract

Let F = {G_{l},...,G_{t}} 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 G_{i} that belong to D. The subset D is called a connected (F,k)-core if the subgraph induced by D in each G_{i} is connected. Let δ_{i} be the minimum degree of G_{i} and let δ(F) = min^{t}_{i=1} δ_{i}. 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 G_{i} is connected. Let c(k,F) and c_{c}(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)), c_{c}(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 language | English |
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Pages (from-to) | 309-313 |

Number of pages | 5 |

Journal | Combinatorics Probability and Computing |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - 2000 |

## ASJC Scopus subject areas

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