On k-rainbow domination in middle graphs
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Let G be a finite simple graph with vertex set V(G) and edge set E(G). A function f : V(G) →𝒫({1, 2, …, k}) is a k-rainbow dominating function on G if for each vertex v ∈ V(G) for which f(v)= ∅, it holds that ⋃_u ∈ N(v)f(u) = {1, 2, …, k}. The weight of a k-rainbow dominating function is the value ∑_v ∈ V(G)|f(v)|. The k-rainbow domination number γ_rk(G) is the minimum weight of a k-rainbow dominating function on G. In this paper, we initiate the study of k-rainbow domination numbers in middle graphs. We define the concept of a middle k-rainbow dominating function, obtain some bounds related to it and determine the middle 3-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle 3-rainbow domination number of trees in terms of the matching number. In addition, we determine the 3-rainbow domatic number for the middle graph of paths and cycles.
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