A function f that assigns values from the set \(\{0, 1, 2\}\) to each vertex of a graph G is called a 2-rainbow independent dominating function, if the vertices assigned the value 1 form an independent set, the vertices assigned the value 2 form another independent set, and every vertex to which 0 is assigned has at least one neighbor in each of the mentioned independent sets. The weight of this function is the total number of vertices assigned nonzero values. The 2-rainbow independent domination number of G, \(\gamma _{\textrm{ri}2}(G)\) , is the minimum weight of such a function. Motivated by a real-life application, we study the 2-rainbow independent domination number of the complementary prism \(G \overline{G}\) of a graph G, which is constructed by taking G and its complement \(\overline{G}\) , and then adding edges between corresponding vertices. We provide tight bounds for \(\gamma _{\textrm{ri}2}(G\overline{G})\) , and characterize graphs for which the lower bound, i.e. \(\max \{\gamma _{\textrm{ri}2}(G), \gamma _{\textrm{ri}2}(\overline{G})\}+1\) , is attained. The obtained results can, in practice, enable the prediction of the cost estimate for a given communication or surveillance network.
