Assume we have a set of k colors and to each vertex of a graph G we assign an arbitrary subset of these colors. If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this is called the k-rainbow dominating function of a graph G. The corresponding invariant γrk(G), which is the minimum sum of numbers of assigned colors over all vertices of G,...

The concept of 2-rainbow domination of a graph G coincides with the ordinary domination of the prism G K2. In this paper, we show that the problem of deciding if a graph has a 2-rainbow dominating function of a given weight is NP-complete even when restricted to bipartite graphs or chordal graphs. Exact values of 2-rainbow domination numbers of several classes of graphs are found, and it is sho...

a {em 2-rainbow dominating function} (2rdf) of 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 for any vertex $vin v(g)$ with $f(v)=emptyset$ the condition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled, where $n(v)$ is the open neighborhood of $v$. the {em weight} of a 2rdf $f$ is the value $omega(f)=sum_{vin v}|f (v)|$. the {em $2$-r...

The rainbow game domination subdivision number of a graph G is defined by the following game. Two players D and A, D playing first, alternately mark or subdivide an edge of G which is not yet marked nor subdivided. The game ends when all the edges of G are marked or subdivided and results in a new graph G′. The purpose of D is to minimize the 2-rainbow dominating number γr2(G ′) of G′ while A t...

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for infinite families, exact values are established; in all other the lower and upper bounds with small gaps given. also define singleton rainbow domination, where sets assigned have a cardinality of, at most, one, provide analogous this special case domination.

We obtain new results on 2-rainbow domination number of generalized Petersen graphs P(5k,k). In some cases (for infinite families), exact values are established, and in all other lower upper bounds given. particular, it is shown that, for k>3, γr2(P(5k,k))=4k k≡2,8mod10, γr2(P(5k,k))=4k+1 k≡5,9mod10, 4k+1≤γr2(P(5k,k))≤4k+2 k≡1,6,7mod10, 4k+1≤γr2(P(5k,k))≤4k+3 k≡0,3,4mod10.

Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the set of in-neighbors of v. The weight of a kRDF f is the value ω(f) = ∑ v∈V |f(v...