Opuscula Mathematica
Opuscula Math. 40, no. 5 (2020), 617-627

On 2-rainbow domination number of functigraph and its complement

Athena Shaminezhad
Ebrahim Vatandoost
Abstract. Let \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp.
Keywords: 2-rainbow domination number, functigraph, complement, cubic graph.
Mathematics Subject Classification: 05C69, 05C75.
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About the authors ▼
Athena Shaminezhad
Imam Khomeini International University, Department of Basic Science, Nouroozian Street, Qazvin, Iran
Ebrahim Vatandoost (corresponding author)
Imam Khomeini International University, Department of Basic Science, Nouroozian Street, Qazvin, Iran
About the article ▼
Communicated by Dalibor Fronček.
Received: 2020-01-06.
Revised: 2020-06-04.
Accepted: 2020-07-07.
Published online: 2020-10-10.
Cite this article as ▼
Athena Shaminezhad, Ebrahim Vatandoost, On 2-rainbow domination number of functigraph and its complement, Opuscula Math. 40, no. 5 (2020), 617-627, https://doi.org/10.7494/OpMath.2020.40.5.617.
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