Fair dominating sets of paths

Abstract

Let G = (V, E)  be a simple graph. A dominating set of G is a subset D ⊆ V such that every vertex not in D is adjacent to at least one vertex in D. The cardinality of the smallest dominating set of G, denoted by g(G),  is the domination number of G. For i ≥ 1,  a i-fair dominating set (iFD-set) in G, is a dominating set S such that |N(v) ∩ D| = i for every vertex  v ∈ V\D. A fair dominating set, in G is a iFD-set for some integer i ≥ 1.  In this paper, we present the structure of fair dominating sets of a path and also we count the number of these sets.