## Abstract

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [Opuscula Math. 31 (2011), 519-531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of the set 5 has no isolated vertex. The neighborhood total domination number, denoted by γnt(G), is the minimum cardinality of a NTD-set of G. Every total dominating set is a NTD-set, implying that γ (G) < γnt (G) ≤ γt(G), where γ (G) and γt(G) denote the domination and total domination numbers of G, respectively. Arumugam and Sivagnanam showed that if G is a connected graph on n vertices with maximum degree < n - 1, then γnt(G) ≤ n - and pose the problem of determining the graphs G achieving equality in this bound. We provide a complete solution to this problem for triangle-free graphs. Further, we give a description involving the packing number of general graphs that achieve equality in the bound.

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

Pages (from-to) | 137-150 |

Number of pages | 14 |

Journal | Utilitas Mathematica |

Volume | 107 |

Publication status | Published - Jun 2018 |

## Keywords

- Domination
- Neighborhood total domination
- Total domination

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics