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Rainbow edge domination numbers in graphs. (English) Zbl 1427.05154

Summary: A 2-rainbow edge dominating function (2REDF) 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 \(e \in E(G)\) with \(f(e) = \emptyset\) the condition \(\bigcup_{e^\prime \in N(e)} f(e^\prime) = \{1, 2 \}\) is fulfilled, where \(N(e)\) is the open neighborhood of \(e\). The weight of a 2REDF \(f\) is the value \(\sum_{e \in E(G)} | f(e) |\). The minimum weight of a 2REDF is the 2-rainbow edge domination number of \(G\), denoted by \(\gamma_{2 r}^\prime(G)\). In this paper, we initiate the study of 2-rainbow edge domination in graphs. We present various sharp bounds, exact values and characterizations for the 2-rainbow edge domination number of a graph.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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