Outer-Connected Fair 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). A set S⊆V(G) is said to be an outer-connected dominating set in G if S is dominating and either S=V(G) or ⟨V(G)∖S⟩ is connected. The outer-connected domination number of G is the minimum cardinality of an outer-connected dominating set of G, denoted by γ ̃_c (G). A fair dominating set in graph G is a dominating set S such that all vertices in V(G)∖S are dominated by the equal number of vertices in S. The fair domination number of G is the minimum cardinality of a fair dominating set of G, denoted by γ_fd (G). A nonempty subset S⊆V (G) is an outer-connected fair dominating set of G, if S is a fair dominating set of G and the subgraph ⟨V(G)∖S⟩ induced by V(G)∖S is connected. The outer-connected fair domination number of G is the minimum cardinality of an outer-connected fair dominating set of G, denoted by γ ̃_cfd (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 ̃_cfd (G) = k for all positive integer k. Further, give the outer-connected fair domination number of some special graphs.