## Abstract

Let G=(V,E) be a graph. A set of vertices 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. The total restrained domination number of G, denoted by γ _{tr} (G), is the smallest cardinality of a total restrained dominating set of G. A support vertex of a graph is a vertex of degree at least two which is adjacent to a leaf. We show that γ_{tr}(T)n+2s+\ell-1}{2} where T is a tree of order n≥3, and s and 〈 are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.

Original language | English |
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Pages (from-to) | 205-223 |

Number of pages | 19 |

Journal | Journal of Combinatorial Optimization |

Volume | 20 |

Issue number | 3 |

DOIs | |

Publication status | Published - Oct 2010 |

## Keywords

- Domination
- Restrained
- Total
- Trees

## ASJC Scopus subject areas

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics