A sharp upper bound on the independent 2-rainbow domination in graphs with minimum degree at least two

摘要

An independent 2-rainbow dominating function (I2-RDF) on a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1,2} such that {x is an element of V vertical bar f(x) = empty set} is an independent set of G and for any vertex upsilon is an element of V(G) with f(upsilon) = empty set we have boolean OR(u is an element of N(upsilon)) f(u) = {1,2}. The weight of an I2-RDF f is the value omega(f) = Sigma(upsilon is an element of V) vertical bar f(upsilon)vertical bar, and the independent 2-rainbow domination number i(r2)(G) is the minimum weight of an I2-RDF on G. In this paper, we prove that if G is a graph of order n = 3 with minimum degree at least two such that the set of vertices of degree at least 3 is independent, then i(r2)(G) = 4n/5.