## 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 of V - S is adjacent to a vertex in V - S. A set S ⊆ V is a restrained dominating set if every vertex in V - S is adjacent to a vertex in S and to a vertex in V - S. The total restrained domination number of G (restrained domination number of G, respectively), denoted by γ_{tr} (G) (γ_{r} (G), respectively), is the smallest cardinality of a total restrained dominating set (restrained dominating set, respectively) of G. We bound the sum of the total restrained domination numbers of a graph and its complement, and provide characterizations of the extremal graphs achieving these bounds. It is known (see [G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar, L.R. Markus, Restrained domination in graphs, Discrete Math. 203 (1999) 61-69.]) that if G is a graph of order n ≥ 2 such that both G and over(G, -) are not isomorphic to P_{3}, then 4 ≤ γ_{r} (G) + γ_{r} (over(G, -)) ≤ n + 2. We also provide characterizations of the extremal graphs G of order n achieving these bounds.

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

Pages (from-to) | 1080-1087 |

Number of pages | 8 |

Journal | Discrete Mathematics |

Volume | 308 |

Issue number | 7 |

DOIs | |

Publication status | Published - 6 Apr 2008 |

## Keywords

- Domination
- Nordhaus-Gaddum
- Restrained
- Total

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics