## Abstract

Let γ(G), ra(G) and ir(G) denote the domination, R-annihilation and irredundance numbers of a graph G, respectively. Graphs whose blocks are claw-free are called script C sign ℱ ℬ-graph. In this paper we establish the best possible upper bounds on the ratios γ(G)/ra(G) and γ(G)/ir(G) in the class of script C sign ℱ ℬ-graphs. The script C sign ℱ ℬ-graphs generalize several classes of graphs for which such ratios have already been investigated. Motivated by our proof methods, we are led to introduce a new family of domination parameters simultaneously generalizing the total domination and k-domination numbers. For two integers, l ≥ 0 and k > 0 a set X of vertices of a graph G=(V, E) is an l-total k-dominating set of G, if every vertex in X has at least l neighbors in X and every vertex in V \ X has at least k neighbors in X. If (at least) one l-total k-dominating set exists, then the l-total k-dominating number γ_{l,k}(G) is the minimum cardinality of such a set. We prove a best possible upper bound on the ratio γ(G)/γ_{1,2}(G) in the class of script C sign ℱ ℬ-graphs. Our bounds on γ(G)/ra(G) and γ(G)/ir(G) for a script C sign ℱ ℬ-graph G will follow as an application of this result.

Original language | English |
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Pages (from-to) | 143-151 |

Number of pages | 9 |

Journal | Discrete Mathematics |

Volume | 231 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 28 Mar 2001 |

Externally published | Yes |

## Keywords

- Annihilation
- Claw
- Domination
- Graph

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics