### Abstract

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_{k} (G), the connected k-domination number γ_{k}^{c} (G); the k-independent domination number γ_{k}^{i} (G) and the k-irredundance number ir_{k} (G). The authors prove that if an ir_{k}-set X is a k-independent set of G, then ir_{k} (G) = γ_{k} (G) = γ_{k}^{i} (G), and that for k ≥ 2, γ_{k}^{c} (G) = 1 if ir_{k} (G) = 1, γ_{k}^{c} (G) ≤ max { (2 k + 1) ir_{k} (G) - 2 k, frac(5, 2) ir_{k} (G) k - frac(7, 2) k + 2 } if ir_{k} (G) is odd, and γ_{k}^{c} (G) ≤ frac(5, 2) ir_{k} (G) k - 3 k + 2 if ir_{k} (G) is even, which generalize some known results.

Original language | English (US) |
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Pages (from-to) | 2943-2953 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 306 |

Issue number | 22 |

DOIs | |

State | Published - Nov 28 2006 |

### Fingerprint

### Keywords

- Connected k-domination number
- k-Domination number
- k-Independent domination number
- k-Independent set
- k-Irredundance number

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*306*(22), 2943-2953. https://doi.org/10.1016/j.disc.2006.03.075

**Distance irredundance and connected domination numbers of a graph.** / Xu, Jun Ming; Tian, Fang; Huang, Jia.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 306, no. 22, pp. 2943-2953. https://doi.org/10.1016/j.disc.2006.03.075

}

TY - JOUR

T1 - Distance irredundance and connected domination numbers of a graph

AU - Xu, Jun Ming

AU - Tian, Fang

AU - Huang, Jia

PY - 2006/11/28

Y1 - 2006/11/28

N2 - Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γk (G), the connected k-domination number γkc (G); the k-independent domination number γki (G) and the k-irredundance number irk (G). The authors prove that if an irk-set X is a k-independent set of G, then irk (G) = γk (G) = γki (G), and that for k ≥ 2, γkc (G) = 1 if irk (G) = 1, γkc (G) ≤ max { (2 k + 1) irk (G) - 2 k, frac(5, 2) irk (G) k - frac(7, 2) k + 2 } if irk (G) is odd, and γkc (G) ≤ frac(5, 2) irk (G) k - 3 k + 2 if irk (G) is even, which generalize some known results.

AB - Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γk (G), the connected k-domination number γkc (G); the k-independent domination number γki (G) and the k-irredundance number irk (G). The authors prove that if an irk-set X is a k-independent set of G, then irk (G) = γk (G) = γki (G), and that for k ≥ 2, γkc (G) = 1 if irk (G) = 1, γkc (G) ≤ max { (2 k + 1) irk (G) - 2 k, frac(5, 2) irk (G) k - frac(7, 2) k + 2 } if irk (G) is odd, and γkc (G) ≤ frac(5, 2) irk (G) k - 3 k + 2 if irk (G) is even, which generalize some known results.

KW - Connected k-domination number

KW - k-Domination number

KW - k-Independent domination number

KW - k-Independent set

KW - k-Irredundance number

UR - http://www.scopus.com/inward/record.url?scp=33749233798&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749233798&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2006.03.075

DO - 10.1016/j.disc.2006.03.075

M3 - Article

AN - SCOPUS:33749233798

VL - 306

SP - 2943

EP - 2953

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 22

ER -