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

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.

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-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...

For a graph G, its k-rainbow independent domination number, written as γrik(G), is defined the cardinality of minimum set consisting k vertex-disjoint sets V1,V2,…,Vk such that every vertex in V0=V(G)\(∪i=1kVi) has neighbor Vi for all i∈{1,2,…,k}. This invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics Computation 333(15), 2018: 353–361), which aims to compute num...

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.