## Abstract

Let n≥1 be an integer and let G be a graph of order p. A set script D sign of vertices of G is a total n-dominating set of G if every vertex of V(G) is within distance n from some vertex of script D sign other than itself. The minimum cardinality among all total n-dominating sets of G is called the total n-domination number and is denoted by γ^{t}_{n}(G). A set S of vertices of G is n-independent if the distance (in G) between every pair of distinct vertices of S is at least n + 1. The minimum cardinality among all maximal n-independent sets of G is called the n-independence number of G and is denoted by i_{n}(G). In this paper, we present an algorithm for finding a total n-dominating set script D sign and a maximal n-independent set S in a connected graph with at least p≥2n+1 vertices. It is shown that these sets script D sign and S satisfy the inequality |S| + n|script D sign|≤p. Using this result, we conclude that if G is a connected graph on p≥2n+1 vertices, then i_{n}(G) + n · γ^{t}_{n}(G)≤p.

Original language | English |
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Pages (from-to) | 85-91 |

Number of pages | 7 |

Journal | Discrete Applied Mathematics |

Volume | 68 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 12 Jun 1996 |

Externally published | Yes |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics