Detour domination in graphs

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 is an element of V (G), define D- (v) = min{D(u,v) : u is an element of V(G) - {v}}. A vertex u (not equal v) is called a detour neighbor of v if D (u, v) = D - (v). A vertex v 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 gamma(D)(G). We show that if G is a connected graph of order n greater than or equal to 3, then gamma(D) (G) less than or equal to n - 2. Moreover, for every pair k, n of integers with I less than or equal to k less than or equal to n - 2, there exists a connected graph G of order n such that gamma(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 gamma(G) = a and gamma(D) (G) = b.