Independent 2-rainbow domination in graphs

Mustapha Chellali1, Nader Jafari Rad2
1LAMDA-RO Laboratory, Department of Mathematics University of Blida. B.P. 270, Blida, Algeria.
2Department of Mathematics, Shahrood University of Technology, Shahrood, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5746, Tehran, Iran

Abstract

A \emph{2-rainbow dominating function} of a graph \( G \) is a function \( g \) that assigns to each vertex a set of colors chosen from the set \( \{1, 2\} \) so that for each vertex \( v \) with \( g(v) = \emptyset \) we have \( \cup_{u \in N(v)} g(u) = \{1, 2\} \). The minimum of \( g(V(G)) = \sum_{v \in V(G)} |g(v)| \) over all such functions is called the \emph{2-rainbow domination number} \( \gamma_{r2}(G) \). A 2-rainbow dominating function \( g \) of a graph \( G \) is independent if no two vertices assigned non-empty sets are adjacent. The \emph{independent 2-rainbow domination number} \( i_{r2}(G) \) is the minimum weight of an independent 2-rainbow dominating function of \( G \). In this paper, we study independent 2-rainbow domination in graphs. We present some bounds and relations with other domination parameters.

Keywords: 2-rainbow domination, independent 2-rainbow domi- nation, Roman domination, independent Roman domination.