### Abstract

A subset S of the vertex set of a hypergraph H is called a dominating set of H if for every vertex v not in S there exists u ε S such that u and v are contained in an edge in H. The minimum cardinality of a dominating set in H is called the domination number of H and is denoted by γ(H). A transversal of a hypergraph H is defined to be a subset T of the vertex set such that Ta ∩ E θ for every edge E of H. The transversal number of H, denoted by τ(H), is the minimum number of vertices in a transversal. A hypergraph is of rank k if each of its edges contains at most k vertices. The inequality τ(H)≥γ(H) is valid for every hypergraph H without isolated vertices. In this paper, we investigate the hypergraphs satisfying τ(H)=γ(H), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class Θ2p. Structurally we focus our attention on hypergraphs in which each subhypergraph H′ without isolated vertices fulfills the equality τ(H′)=γ(H′). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer k, there are only a finite number of forbidden subhypergraphs of rank k, and each of them has domination number at most k.

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

Pages (from-to) | 1859-1867 |

Number of pages | 9 |

Journal | Discrete Applied Mathematics |

Volume | 161 |

Issue number | 13-14 |

DOIs | |

Publication status | Published - Sep 2013 |

### Fingerprint

### Keywords

- Computational complexity
- Domination number
- Hereditary property
- Hitting set
- Hypergraph
- Polynomial hierarchy
- Transversal number

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Applied Mathematics*,

*161*(13-14), 1859-1867. https://doi.org/10.1016/j.dam.2013.02.009

**Equality of domination and transversal numbers in hypergraphs.** / Arumugam, S.; Jose, Bibin K.; Bujtás, Csilla; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 161, no. 13-14, pp. 1859-1867. https://doi.org/10.1016/j.dam.2013.02.009

}

TY - JOUR

T1 - Equality of domination and transversal numbers in hypergraphs

AU - Arumugam, S.

AU - Jose, Bibin K.

AU - Bujtás, Csilla

AU - Tuza, Z.

PY - 2013/9

Y1 - 2013/9

N2 - A subset S of the vertex set of a hypergraph H is called a dominating set of H if for every vertex v not in S there exists u ε S such that u and v are contained in an edge in H. The minimum cardinality of a dominating set in H is called the domination number of H and is denoted by γ(H). A transversal of a hypergraph H is defined to be a subset T of the vertex set such that Ta ∩ E θ for every edge E of H. The transversal number of H, denoted by τ(H), is the minimum number of vertices in a transversal. A hypergraph is of rank k if each of its edges contains at most k vertices. The inequality τ(H)≥γ(H) is valid for every hypergraph H without isolated vertices. In this paper, we investigate the hypergraphs satisfying τ(H)=γ(H), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class Θ2p. Structurally we focus our attention on hypergraphs in which each subhypergraph H′ without isolated vertices fulfills the equality τ(H′)=γ(H′). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer k, there are only a finite number of forbidden subhypergraphs of rank k, and each of them has domination number at most k.

AB - A subset S of the vertex set of a hypergraph H is called a dominating set of H if for every vertex v not in S there exists u ε S such that u and v are contained in an edge in H. The minimum cardinality of a dominating set in H is called the domination number of H and is denoted by γ(H). A transversal of a hypergraph H is defined to be a subset T of the vertex set such that Ta ∩ E θ for every edge E of H. The transversal number of H, denoted by τ(H), is the minimum number of vertices in a transversal. A hypergraph is of rank k if each of its edges contains at most k vertices. The inequality τ(H)≥γ(H) is valid for every hypergraph H without isolated vertices. In this paper, we investigate the hypergraphs satisfying τ(H)=γ(H), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class Θ2p. Structurally we focus our attention on hypergraphs in which each subhypergraph H′ without isolated vertices fulfills the equality τ(H′)=γ(H′). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer k, there are only a finite number of forbidden subhypergraphs of rank k, and each of them has domination number at most k.

KW - Computational complexity

KW - Domination number

KW - Hereditary property

KW - Hitting set

KW - Hypergraph

KW - Polynomial hierarchy

KW - Transversal number

UR - http://www.scopus.com/inward/record.url?scp=84878287398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878287398&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2013.02.009

DO - 10.1016/j.dam.2013.02.009

M3 - Article

VL - 161

SP - 1859

EP - 1867

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 13-14

ER -