2-Rainbow domination stability of graphs

Abstract

For a graph G, let $$f:V(G)\rightarrow \mathcal {P}(\{1,2\}).$$ If for each vertex $$v\in V(G)$$ such that $$f(v)=\emptyset $$ we have $$\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2\},$$ then f is called a 2-rainbow dominating function (2RDF) of G. The weight w(f) of a function f is defined as $$w(f)=\sum _{v\in V(G)}\left| f(v)\right| $$ . The minimum weight of a 2RDF of G is called the 2-rainbow domination number of G, denoted by $$\gamma _{r2}(G)$$ . The 2-rainbow domination stability, $$st_{\gamma r2}(G)$$ , 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 $$st_{\gamma r2}(G)$$ . In particular, we obtain $$st_{\gamma r2}(G)\le \delta (G)+1$$ and $$st_{\gamma r2}(G)\le |V(G)|-\varDelta (G)-1$$ if $$\gamma _{r2}(G)\ge 3$$ . Moreover, we prove that there exists no graph G with $$st_{\gamma r2}(G)=|V(G)|-2$$ when $$n\ge 4$$ and characterize the graphs G with $$st_{\gamma r2}(G)=|V(G)|-1$$ or $$st_{\gamma r2}(G)=|V(G)|-3$$ .