Fair Detour Domination of Graphs

Abstract

A set [Formula: see text] of a connected graph [Formula: see text] 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, [Formula: see text], of G is the minimum cardinality of fair detour dominating sets. A fair detour dominating set of cardinality [Formula: see text] is called a [Formula: see text]-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 [Formula: see text] and [Formula: see text] then [Formula: see text]. It is shown that for given positive integers [Formula: see text], [Formula: see text], [Formula: see text] such that [Formula: see text] there exists a connected graph G of order [Formula: see text] such that [Formula: see text] and [Formula: see text].