Opuscula Math. 36, no. 5 (2016), 563-574

Opuscula Mathematica

Criticality indices of 2-rainbow domination of paths and cycles

Ahmed Bouchou
Mostafa Blidia

Abstract. A \(2\)-rainbow dominating function of a graph \(G\left(V(G),E(G)\right)\) is a function \(f\) that assigns to each vertex a set of colors chosen from the set \(\{1,2\}\) so that for each vertex with \(f(v)=\emptyset\) we have \({\textstyle\bigcup_{u\in N(v)}} f(u)=\{1,2\}\). The weight of a \(2\)RDF \(f\) is defined as \(w\left( f\right)={\textstyle\sum\nolimits_{v\in V(G)}} |f(v)|\). The minimum weight of a \(2\)RDF is called the \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{2r}(G)\). The vertex criticality index of a \(2\)-rainbow domination of a graph \(G\) is defined as \(ci_{2r}^{v}(G)=(\sum\nolimits_{v\in V(G)}(\gamma_{2r}\left(G\right) -\gamma_{2r}\left( G-v\right)))/\left\vert V(G)\right\vert\), the edge removal criticality index of a \(2\)-rainbow domination of a graph \(G\) is defined as \(ci_{2r}^{-e}(G)=(\sum\nolimits_{e\in E(G)}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G-e\right)))/\left\vert E(G)\right\vert\) and the edge addition of a \(2\)-rainbow domination criticality index of \(G\) is defined as \(ci_{2r}^{+e}(G)=(\sum\nolimits_{e\in E(\overline{G})}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G+e\right)))/\left\vert E(\overline{G})\right\vert\), where \(\overline{G}\) is the complement graph of \(G\). In this paper, we determine the criticality indices of paths and cycles.

Keywords: 2-rainbow domination number, criticality index.

Mathematics Subject Classification: 05C69.

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  • Ahmed Bouchou
  • University of Médéa, Algeria
  • Mostafa Blidia
  • University of Blida, LAMDA-RO, Department of Mathematics, B.P. 270, Blida, Algeria
  • Communicated by Hao Li.
  • Received: 2014-12-06.
  • Revised: 2015-02-15.
  • Accepted: 2016-03-02.
  • Published online: 2016-06-29.
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Cite this article as:
Ahmed Bouchou, Mostafa Blidia, Criticality indices of 2-rainbow domination of paths and cycles, Opuscula Math. 36, no. 5 (2016), 563-574, http://dx.doi.org/10.7494/OpMath.2016.36.5.563

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