### Abstract

A k-dominating set in a graph G is a set S of vertices such that every vertex of G is at distance at most k from some vertex of S. Given a class D of finite simple graphs closed under connected induced subgraphs, we completely characterize those graphs G in which every connected induced subgraph has a connected k-dominating subgraph isomorphic to some D∈D. We apply this result to prove that the class of graphs hereditarily D-dominated within distance k is the same as the one obtained by iteratively taking the class of graphs hereditarily dominated by the previous class in the iteration chain. This strong relation does not remain valid if the initial hereditary restriction on D is dropped.

Original language | English |
---|---|

Pages (from-to) | 2672-2675 |

Number of pages | 4 |

Journal | Discrete Mathematics |

Volume | 312 |

Issue number | 17 |

DOIs | |

Publication status | Published - Sep 6 2012 |

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### Keywords

- Class of connected graphs
- Distance domination
- Dominating set
- Forbidden induced subgraph

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*312*(17), 2672-2675. https://doi.org/10.1016/j.disc.2011.12.008

**Distance domination versus iterated domination.** / Bacsó, Gábor; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 312, no. 17, pp. 2672-2675. https://doi.org/10.1016/j.disc.2011.12.008

}

TY - JOUR

T1 - Distance domination versus iterated domination

AU - Bacsó, Gábor

AU - Tuza, Z.

PY - 2012/9/6

Y1 - 2012/9/6

N2 - A k-dominating set in a graph G is a set S of vertices such that every vertex of G is at distance at most k from some vertex of S. Given a class D of finite simple graphs closed under connected induced subgraphs, we completely characterize those graphs G in which every connected induced subgraph has a connected k-dominating subgraph isomorphic to some D∈D. We apply this result to prove that the class of graphs hereditarily D-dominated within distance k is the same as the one obtained by iteratively taking the class of graphs hereditarily dominated by the previous class in the iteration chain. This strong relation does not remain valid if the initial hereditary restriction on D is dropped.

AB - A k-dominating set in a graph G is a set S of vertices such that every vertex of G is at distance at most k from some vertex of S. Given a class D of finite simple graphs closed under connected induced subgraphs, we completely characterize those graphs G in which every connected induced subgraph has a connected k-dominating subgraph isomorphic to some D∈D. We apply this result to prove that the class of graphs hereditarily D-dominated within distance k is the same as the one obtained by iteratively taking the class of graphs hereditarily dominated by the previous class in the iteration chain. This strong relation does not remain valid if the initial hereditary restriction on D is dropped.

KW - Class of connected graphs

KW - Distance domination

KW - Dominating set

KW - Forbidden induced subgraph

UR - http://www.scopus.com/inward/record.url?scp=84862691000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862691000&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2011.12.008

DO - 10.1016/j.disc.2011.12.008

M3 - Article

VL - 312

SP - 2672

EP - 2675

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 17

ER -