Fair detour domination of graphs

A set D subset of V of a connected graph G=(V,E) is called a fair detour dominating set if D is a detour dominating set and every two vertices not in D has same number of neighbors in D. The fair detour domination number, f gamma d, of G is the minimum cardinality of fair detour dominating sets. A fair detour dominating set of cardinality f gamma d is called a f gamma d-set of G. The fair detour domination number of some well-known graphs are determined. We have shown that, If G is a connected graph with p >= 4 and delta >= 2 then f gamma d(G) <= p -2. It is shown that for given positive integers p >= 4, k, m such that 2 <= k <= m <= p -2 there exists a connected graph G of order p such that gamma d=k and f gamma d=m.