Outer independent global dominating set of trees and unicyclic graphs - Doost Ali Mojdeh; Mortaza Alishahi (Electronic Journal of Graph Theory and Applications (EJGTA), 2019)
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 $\overline G$. γg(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 γgoi(G), is called the outer independent global domination number of G. An outer independent global dominating set of cardinality γgoi(G) is called a γgoi-set of G. In this paper we characterize trees T for which γgoi(T) = γ(T) and trees T for which γgoi(T) = γg(T) and trees T for which γgoi(T) = γoi(T) and the unicyclic graphs G for which γgoi(G) = γ(G), and the unicyclic graphs G for which γgoi(G) = γg(G).