## Abstract

A graph is called γ-critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γ_{t}-critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γ_{t}-critical graph that has total domination number k, is called k-γ_{t}-critical. In this paper, we introduce a class of k-γ_{t}-critical graphs of high connectivity for each integer k ≥ 3. In particular, we provide a partial answer to the question "Which graphs are γ-critical and γ_{t}-critical or one but not the other?" posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255-261].

Original language | English |
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Pages (from-to) | 1969-1973 |

Number of pages | 5 |

Journal | Discrete Applied Mathematics |

Volume | 157 |

Issue number | 8 |

DOIs | |

Publication status | Published - 28 Apr 2009 |

Externally published | Yes |

## Keywords

- Connectivity
- Diameter
- Total domination
- Vertex critical

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics