ON THE INDEPENDENT RAINBOW DOMINATION STABLE GRAPHS

Abstract

"For a graph G and an integer k ≥ 2, let f : V (G) → P({1, 2, ..., k}) be a function. If for each vertex v ∈ V (G) such that f(v) = ∅ we have ∪u∈N(v)f(u) = {1, 2, ..., k}, then f is called a k-rainbow dominating function (or simply kRDF) of G. The weight of a kRDF f is defined as w(f) = P v∈V (G) |f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, and is denoted by γrk(G). An independent k-rainbow dominating function (IkRDF) is a kRDF f with the property that {v : f(v) ̸= ∅} is an independent set. The minimum weight of an IkRDF of G is called the independent k-rainbow domination number of G, and is denoted by irk(G). A graph G is k-rainbow domination stable if the k-rainbow domination number of G remains unchanged under removal of any vertex. Likewise, a graph G is independent k-rainbow domination stable if the independent k-rainbow domination number of G remains unchanged under removal of any vertex. In this paper, we prove that determining whether a graph is k-rainbow domination stable or independent k-rainbow domination stable is NP-hard even when restricted to bipartite or planar graphs, thus answering a question posed in [11]."