## Abstract

In this paper, we continue the study of total restrained domination in graphs. A set S of vertices in a graph G = (V, E) is a total restrained dominating set of G if every vertex of G is adjacent to some vertex in S and every vertex of V\S is adjacent to a vertex in V\S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number γ _{tr}(G) of G. Jiang et al. (Graphs Combin 25:341-350, 2009) showed that if G is a connected cubic graph of order n, then γ _{tr}(G) ≤ 13n/19. In this paper we improve this upper bound to γ _{tr}(G) ≤ (n + 4)/2. We provide two infinite families of connected cubic graphs G with γ _{tr}(G) = n/2, showing that our new improved bound is essentially best possible.

Original language | English |
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Pages (from-to) | 547-554 |

Number of pages | 8 |

Journal | Graphs and Combinatorics |

Volume | 28 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2012 |

## Keywords

- Bounds
- Cubic graphs
- Total retrained domination

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics