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 any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$‎, ‎we define a‎ ‎function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating‎ ‎function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least‎ ‎$k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$‎. ‎The minimum weight of a Roman $k$-tuple dominatin...

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

A Roman dominating function of a graph G is a function f : V → {0, 1, 2} such that every vertex with 0 has a neighbor with 2. The minimum of f (V (G)) = ∑ v∈V f (v) over all such functions is called the Roman domination number γR(G). A 2-rainbow dominating function of a graphG is a function g that assigns to each vertex a set of colors chosen from the set {1, 2}, for each vertex v ∈ V (G) such ...