Identifying Inverse Domination in Graphs

Let G=(V(G),E(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 S⊆V(G)\D is a dominating set of G, then S is called an inverse dominating set with respect to D. The inverse domination number of G is the minimum cardinality of an inverse dominating set of G, denoted by γ^(-1) (G). An identifying code of a graph G is a dominating set C⊆V(G) such that for every v ∈ V(G), N_G [v] ∩ C is distinct. The minimum cardinality of an identifying code of G, denoted by γ^ID (G), is called the identifying code number of G. An inverse dominating set S⊆V(G)∖D is an identifying inverse dominating set of G if for every v∈V(G), N_G [v] ∩ C is distinct. The minimum cardinality of an identifying inverse dominating set of G, denoted by γ^(ID(-1)) (G), is called the identifying inverse domination number of G. In this paper, we initiate the study of the concept and we show the existence of a connected graph G with |V(G)| = n and γ^(ID(-1)) (G) = k for all positive integer k. Further, we give the identifying inverse domination number of a path graph.