## Abstract

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G. These graphs we call γ_{t}-critical. If such a graph G has total domination number k, we call it k-γ_{t}-critical. We characterize the connected graphs with minimum degree one that are γ _{t}-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-γ_{t}-critical graph for k≤8 and provide an example which shows that the maximum diameter is in general at least 5k/3 - O(1).

Original language | English |
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Pages (from-to) | 255-261 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 286 |

Issue number | 3 |

DOIs | |

Publication status | Published - 28 Sept 2004 |

Externally published | Yes |

## Keywords

- Bounds; Diameter
- Total domination
- Vertex critical

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics