Some Results on the Maximal 2-Rainbow Domination Number in Graphs

Abstract

A 2-rainbow dominating function ( ) of a graph is a function from the vertex set to the set of all subsets of the set such that for any vertex with the condition is fulfilled, where is the open neighborhood of . A maximal 2-rainbow dominating function on a graph is a 2-rainbow dominating function such that the set is not a dominating set of . The weight of a maximal is the value . The maximal 2-rainbow domination number of a graph , denoted by , is the the minimum weight of a maximal of . In this paper, we continue the study of maximal 2-rainbow domination number. We characterize all graphs of order whose maximal 2-rainbow domination number is equal to 2 or 3. Finally, we characterize all graphs of order with for which .