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

Abstract: A 2-rainbow domination function of a graph G = (V, E) is a function f mapping each vertex v to a subset of {1, 2} such that ⋃ u∈N(v) f (u) = {1, 2} when f (v) = �, where N(v) is the open neighborhood of v. The weight of f is denoted by wf (G) = ∑ v∈V �f (v)�. The 2-rainbow domination number, denoted by r2(G), is the smallest wf (G) among all 2-rainbow domination functions f of G. The ...

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.......................

for every positive integer k, a set s of vertices in a graph g = (v;e) is a k- tuple dominating set of g if every vertex of v -s is adjacent to at least k vertices and every vertex of s is adjacent to at least k - 1 vertices in s. the minimum cardinality of a k-tuple dominating set of g is the k-tuple domination number of g. when k = 1, a k-tuple domination number is the well-studied domination...

We obtain new results on the 2-rainbow domination number of generalized Petersen graphs P(ck,k). Exact values are established for all infinite families where general lower bound 45ck is attained. In other cases and upper bounds with small gaps given.

Let f be a function that assigns to each vertex a subset of colors chosen from a set C = {1, 2, . . . , k} of k colors. If  u∈N(v) f (u) = C for each vertex v ∈ V with f (v) = ∅, then f is called a k-rainbow dominating function (kRDF) of G where N(v) = {u ∈ V | uv ∈ E}. The weight of f , denoted by w(f ), is defined as w(f ) =  v∈V |f (v)|. Given a graph G, the minimum weight among all weight...

For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination...

A 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 v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2} is fulfilled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value ω(f) = ∑ v∈V |f(v)|. The 2-rainbow domination number of a graph G, denoted by γr2...