## Abstract

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, Υ_{t}(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, Υ^{d}_{t} (G), is the minimum cardinality of such a set. We observe that Υ^{d}_{t} (G) ≤ Υ_{t}(G). A leaf of G is a vertex of degree 1, while a support vertex of G is a vertex adjacent to a leaf. We show that if T is a tree of order n with . leaves and s support vertices, then 2(n-ℓ+3)/5 ≤ Υ^{d}_{t} (T) . (n+s.1)/2 and we characterize the families of trees which attain these bounds. For every tree T, we show have Υ_{t}(T)/Υ^{d}_{t} (T) ≤ 2 and this bound is asymptotically tight.

Original language | English |
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Pages (from-to) | 153-171 |

Number of pages | 19 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 36 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 |

## Keywords

- Disjunctive total domination
- Total domination
- Trees

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics