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 . ...

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (D) 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. A set {f1, f2, . . . , fd} of k-rainbow dominating functions on D with t...

A edge 2-rainbow dominating function (E2RDF) of a graph G is a ‎function f from the edge set E(G) to the set of all subsets‎ ‎of the set {1,2} such that for any edge.......................

‎Let G be a graph‎. ‎A 2-rainbow dominating function (or‎ 2-RDF) of G is a function f from V(G)‎ ‎to the set of all subsets of the set {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled‎, wher NG(v)  is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎$omega (f):=sum _{vin V(G)}|f(v)|$‎. ‎The 2-rainbow‎‎d...

A k-rainbow dominating function of a graph G is a map f from V (G) to the set of all subsets of {1, 2, . . . , k} such that {1, . . . , k} = ⋃ u∈N(v) f(u) whenever v is a vertex with f(v) = ∅. The k-rainbow domination number of G is the invariant γrk(G), which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function. We focu...

For a positive integer k, a k-rainbow dominating function 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, . . . , k} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1, f2, ....

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} o...

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...

a 2-emph{rainbow dominating function} (2rdf) on a graph $g=(v, e)$ is afunction $f$ from the vertex set $v$ to the set of all subsets of the set${1,2}$ such that for any vertex $vin v$ with $f(v)=emptyset$ thecondition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled. a 2rdf $f$ isindependent (i2rdf) if no two vertices assigned nonempty sets are adjacent.the emph{weight} of a 2rdf $f$ is the value $o...

For a graph G, 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, w(f), of a kRDF f is defined as w(f) = ∑ 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 ...