### Abstract

We solve several conjectures and open problems from a recent paper by ACHARYA [2]. Some of our results are relatives of the Nordhaus-Gaddum theorem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set D X such that, for every vertex x Ie{cyrillic, ukrainian} X\D there exists an edge E Ie{cyrillic, ukrainian} ε with x Ie{cyrillic, ukrainian} E and ) As an example, it is shown that the tight bound γγ(H) plus γγ(H) ≤ n+2 holds in hypergraphs H = (X, ε) of order n ≥ 6, where is defined as with, and γγ is the minimum total cardinality of two disjoint dominating sets. We also present some simple constructions of balanced hypergraphs, disproving conjectures of the aforementioned paper concerning strongly independent sets. (Hypergraph H is balanced if every odd cycle in H has an edge containing three vertices of the cycle; and a set S ⊆ X is strongly independent if ≤ 1 for all E Ie{cyrillic, ukrainian} ε.).

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

Pages (from-to) | 347-358 |

Number of pages | 12 |

Journal | Applicable Analysis and Discrete Mathematics |

Volume | 3 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 1 2009 |

### Keywords

- Balanced hypergraph
- Disjoint domination number
- Independent domination number
- Inverse domination number
- Strongly independent set

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Hypergraph domination and strong independence'. Together they form a unique fingerprint.

## Cite this

*Applicable Analysis and Discrete Mathematics*,

*3*(2), 347-358. https://doi.org/10.2298/AADM0902347J