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 o i ( G ) , is called the outer independent global domination number of G . An outer independent global dominating set of cardinality γ g o i ( G ) is called a γ g o i -set of G . In this paper we characterize trees T for which γ g o i ( T ) = γ ( T ) and trees T for which γ g o i ( T ) = γ g ( T ) and trees T for which γ g o i ( T ) = γ o i ( T ) and the unicyclic graphs G for which γ g o i ( G ) = γ ( G ) , and the unicyclic graphs G for which γ g o i ( G ) = γ g ( G ) .