Critical concept for 2-rainbow domination in graphs

For a graph G, let f : V (G) → P({1, 2, . . . , k}) be a function. If for each vertex v ∈ V (G) such that f(v) = ∅ we have ∪u∈N(v)f(u) = {1, 2, . . . , k}, then f is called a k-rainbow dominating function (or simply kRDF) of G. The weight, w(f), of a kRDF f is defined as w(f) = ∑ v∈V (G) |f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, and is denoted by γkr(G). The concept of criticality with respect to various operations on graphs has been studied for several domination parameters. In this paper we study the concept of criticality for 2-rainbow domination in graphs. We characterize 2-rainbow domination vertex (edge) super critical graphs and we will give several characterizations for 2-rainbow domination vertex (edge) critical graphs.