## Abstract

For distinct vertices u and v of a nontrivial connected graph G, the detour distance D(u, v) between u and v is the length of a longest u-v path in G. For a vertex v ∈ V(G), define D^{-}(v) = min{D(u, v) : u ∈ V(G) - {v}}. A vertex u (≠ v) is called a detour neighbor of v if D(u, v) = D ^{-}(v). A vertex u is said to detour dominate a vertex u if u = v or u is a detour neighbor of v. A set S of vertices of G is called a detour dominating set if every vertex of G is detour dominated by some vertex in S. A detour dominating set of G of minimum cardinality is a minimum detour dominating set and this cardinality is the detour domination number γ_{D}(G) . We show that if G is a connected graph of order n ≥ 3, then γ_{D}(G) ≤ n - 2. Moreover, for every pair k, n of integers with 1 ≤ k ≤ n - 2, there exists a connected graph G of order n such that γ_{D}(G) = k. It is also shown that for each pair a, b of positive integers, there is a connected graph G with domination number γ(G) = a and γ_{D}(G) = b.

Original language | English |
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Pages (from-to) | 149-160 |

Number of pages | 12 |

Journal | Ars Combinatoria |

Volume | 71 |

Publication status | Published - Apr 2004 |

Externally published | Yes |

## Keywords

- Detour distance
- Detour domination

## ASJC Scopus subject areas

- General Mathematics