### 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*).

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.1.10

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