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On 2-rainbow domination number of functigraph and its complement

Shaminezhad Athena, Vatandoost Ebrahim

Opuscula Mathematica

2020

Vol. 40, no. 5

617--627

Let G be a graph and ƒ : V(G) → P({1, 2}) be a function where for every vertex v ∈ V(G), with ƒ (v) = ∅ we have [formula]. Then ƒ is a 2-rainbow dominating function or a 2RDF of G. The weight of ƒ is[formula]. The minimum weight of all 2-rainbow dominating functions is 2-rainbow domination number of G, denoted by [formula]. Let G 1 and G2 be two copies of a graph G with disjoint vertex sets V(G 1) and V(G2), and let σ be a function from V(G 1) to V(G2). We define the functigraph C(G,σ) to be the graph that has the vertex set V(C(G, ,σ)) = V(G 1) U V(G2), and the edge set [formula]. In this paper, 2-rainbow domination number of the functigraph of C(G, ,σ) and its complement are investigated. We obtain a general bound for [formula] and we show that this bound is sharp.

Criticality indices of 2-rainbow domination of paths and cycles

Bouchou A., Blidia M.

Opuscula Mathematica

2016

Vol. 36, no. 5

563-–574

A 2-rainbow dominating function of a graph G (V(G), E(G)) is a function ƒ that assigns to each vertex a set of colors chosen from the set {1,2} so that for each vertex with ƒ (v) = ∅ we have [formula].The weight of a 2RDF ƒ is defined as [formula] minimum weight of a 2RDF is called the 2-rainbow domination number of G, denoted by [formula].The vertex criticality index of a 2-rainbow domination of a graph G is defined as [formula] the edge removal criticality index of a 2-rainbow domination of a graph G is defined as [formula] and the edge addition of a 2-rainbow domination criticality index of G is defined as [formula] where G is the complement graph of G. In this paper, we determine the criticality indices of paths and cycles.