Outer independent global dominating set of trees and unicyclic graphs

Abstract

Let G be a graph. A set D ⊆ V(G) is a global dominating set of G if D is a dominating set of G and $\overline G$. γ_g(G) denotes global domination number of G. A set D ⊆ V(G) is an outer independent global dominating set (OIGDS) of G if D is a global dominating set of G and V(G) − D is an independent set of G. The cardinality of the smallest OIGDS of G, denoted by γ_g^oi(G), is called the outer independent global domination number of G. An outer independent global dominating set of cardinality γ_g^oi(G) is called a γ_g^oi-set of G. In this paper we characterize trees T for which γ_g^oi(T) = γ(T) and trees T for which γ_g^oi(T) = γ_g(T) and trees T for which γ_g^oi(T) = γ^oi(T) and the unicyclic graphs G for which γ_g^oi(G) = γ(G), and the unicyclic graphs G for which γ_g^oi(G) = γ_g(G).