2-Rainbow domination stability of 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 weight w(f) of a function 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, denoted by $$\gamma _{r2}(G)$$ γ r 2 ( G ) . The 2-rainbow domination stability, $$st_{\gamma r2}(G)$$ s t γ r 2 ( G ) , of G is the minimum number of vertices in G whose removal changes the 2-rainbow domination number. In this paper, we first determine the exact values on 2-rainbow domination stability of some special classes of graphs, such as paths, cycles, complete graphs and complete bipartite graphs. Then we obtain several bounds on $$st_{\gamma r2}(G)$$ s t γ r 2 ( G ) . In particular, we obtain $$st_{\gamma r2}(G)\le \delta (G)+1$$ s t γ r 2 ( G ) ≤ δ ( G ) + 1 and $$st_{\gamma r2}(G)\le |V(G)|-\varDelta (G)-1$$ s t γ r 2 ( G ) ≤ | V ( G ) | - Δ ( G ) - 1 if $$\gamma _{r2}(G)\ge 3$$ γ r 2 ( G ) ≥ 3 . Moreover, we prove that there exists no graph G with $$st_{\gamma r2}(G)=|V(G)|-2$$ s t γ r 2 ( G ) = | V ( G ) | - 2 when $$n\ge 4$$ n ≥ 4 and characterize the graphs G with $$st_{\gamma r2}(G)=|V(G)|-1$$ s t γ r 2 ( G ) = | V ( G ) | - 1 or $$st_{\gamma r2}(G)=|V(G)|-3$$ s t γ r 2 ( G ) = | V ( G ) | - 3 .