## Abstract

Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S, and every vertex in V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted γ _{tr} (G), is the smallest cardinality of a total restrained dominating set of G. In Koh et al. (2013), it is proved that if G is a graph of order n≥4 and δ(G)≥2, then γ _{tr} (G)≤n−[Formula presented]3. It is further conjectured that this bound can be improved to γ _{tr} (G)≤n−θ(n). In this paper we show that if G is a graph with no C _{3} components and δ(G)≥2, then γ _{tr} (G)≤n−[Formula presented].

Original language | English |
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Pages (from-to) | 177-187 |

Number of pages | 11 |

Journal | Discrete Applied Mathematics |

Volume | 258 |

DOIs | |

Publication status | Published - 15 Apr 2019 |

## Keywords

- Domination
- Graph
- Total restrained domination
- Upper bound

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics