# Outer independent Roman domination number of trees

Document Type : Original paper

Authors

1 Sirjan University of Technology, Sirjan 78137, Iran

2 LAMDA-RO Laboratory, Department of Mathematics University of Blida B.P. 270, Blida, Algeria

Abstract

A Roman dominating function (RDF) on a graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. An RDF $f$ is called an outer independent Roman dominating function (OIRDF) if the set of vertices assigned a $0$ under $f$ is an independent set. The weight of an OIRDF is the sum of its function values over all vertices, and the outer independent Roman domination number $\gamma _{oiR}(G)$ is the minimum weight of an OIRDF on $G$. In this paper, we show that if $T$ is a tree of order $n\geq 3$ with $s(T)$ support vertices, then $\gamma _{oiR}(T)\leq \min \{\frac{5n}{6},\frac{3n+s(T)}{4}\}.$ Moreover, we characterize the tress attaining each bound.

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