On the 2-rainbow domination stable graphs

For a graph G, let $$f:V(G)\rightarrow {\mathcal {P}}(\{1,2\}).$$ f : V ( G ) → P ( { 1 , 2 } ) . If for each vertex $$v\in V(G)$$ v ∈ V ( G ) such that $$f(v)=\emptyset $$ f ( v ) = ∅ we have $$\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2\}, $$ ⋃ u ∈ N ( v ) f ( u ) = { 1 , 2 } , then f is called a 2-rainbow dominating function (2RDF) of G. The weightw(f) of f is defined as $$w(f)=\sum _{v\in V(G)}\left| f(v)\right| $$ w ( f ) = ∑ v ∈ V ( G ) f ( v ) . The minimum weight of a 2RDF of G is called the 2-rainbow domination number of G, which is denoted by $$\gamma _{r2}(G)$$ γ r 2 ( G ) . A graph G is 2-rainbow domination stable if the 2-rainbow domination number of G remains unchanged under removal of any vertex. In this paper, we prove that determining whether a graph is 2-rainbow domination stable is NP-hard and characterize 2-rainbow domination stable trees.