## Abstract

A set S of vertices in a graph G is a total dominating set of G if every vertex has a neighbor in S. The total domination number, γ_{t}(G), is the minimum cardinality of a total dominating set of G. Let n_{G} and m_{G} denote the number of vertices and edges, respectively, in G. We prove that all the essential upper bounds on the total domination number of a graph G without isolated vertices and isolated edges can be written in the unified form γ_{t}(G)≤(2an_{G}+2bm_{G})∕(3a+2b) for constants b≥0 and a≥[Formula presented](1−b). We also provide unified forms of upper bounds for the total domination number of a graph with minimum degree δ in the cases when δ∈{2,3,4}. For example, we show that the bound γ_{t}(G)≤an_{G}+bm_{G} is valid for every graph G with minimum degree δ=3 if and only if both b≥0 and a≥[Formula presented](1−3b) hold.

Original language | English |
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Pages (from-to) | 103-115 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 244 |

DOIs | |

Publication status | Published - 31 Jul 2018 |

## Keywords

- Total domination
- Upper bounds

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics