On the 2-rainbow domination stable graphs

Zepeng Li, Zehui Shao, Pu Wu, Taiyin Zhao

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 weightw(f) of 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, which is denoted by \(\gamma _{r2}(G)\). 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.

External IDs:dblp:journals/jco/LiSWZ19