Fair Inverse Domination in Graphs

Let G be a nontrivial connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈ V(G)\S, there exists x∈S such that xv∈E(G). Let D be a minimum dominating set of G. If V(G)\D contains a dominating set say S of G, then S is called an inverse dominating set with respect to D. A fair dominating set in a graph G (or FD-set) is a dominating set S such that all vertices not in S are dominated by the same number of vertices from S ; that is, every two vertices not in S has the same number of neighbors in S. An inverse dominating subset S of a vertex set V(G) is said to be fair inverse dominating set if for every vertex v∈ V(G)\S is dominated by the same number of the vertex in S. A fair inverse domination number is the minimum cardinality of a fair inverse dominating set S in G. In this paper, we initiate the study of the concept and give the fair inverse domination number of some special graphs. Further, we give the characterization of the fair inverse dominating set in the join of two nontrivial connected graphs.