Amjadi, J.; Parnian, A.; Sheikholeslami, S. M.; Dehgardi, N.; Volkmann, L. Mixed \(k\)-rainbow domination numbers in graphs. (English) Zbl 1464.05278 Util. Math. 108, 53-71 (2018). Let \(G=(V,E)\) be a simple graph. A dominating set of \(G\) is a subset \(S\subseteq V\) such that every vertex not in \(S\) is adjacent to at least one vertex in \(S\). The cardinality of a smallest dominating set of \(G\), denoted by \(\gamma(G)\), is the domination number of \(G\). A set \(S\subseteq V\cup E\) such that every element not in \(S\) is adjacent or incident to at least one element in \(S\) is called mixed dominating set of \(G\). The cardinality of a smallest mixed dominating set of \(G\), denoted by \(\gamma_m(G)\), is the mixed domination number of \(G\). Let \(k\) be a positive integer, and set \([k]:=\{1,2,\dots,k\}\). For a graph \(G\), a \(k\)-rainbow dominating function (or kRDF) of \(G\) is a mapping \(f:V(G)\rightarrow 2^{[k]}=P([k])\) in such a way that, for any vertex \(v\in V(G)\) with \(f(v)=\emptyset\), the condition \(\cup_{u\in N(v)}f(u)=[k]\) always holds, where \(N(v)\) is the open neighborhood of \(v\). The weight of kRDF \(f\) of \(G\) is the summation of values of all vertices under \(f\). The \(k\)-rainbow domination number of \(G\), denoted by \(γ_{rk}(G)\), is the minimum weight of a kRDF of \(G\). Similar to the definition of mixed domination number, the authors extended the definition of \(k\)-rainbow domination number to the mixed \(k\)-rainbow domination number of \(G\) which is denoted by \(γ_{rk}^m(G)\). Authors in this paper presented some sharp bounds for the mixed \(k\)-rainbow domination number. Also they computed the mixed \(2\)-rainbow domination number of some specific graphs. Reviewer: Saeid Alikhani (Yazd) MSC: 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) Keywords:mixed \(k\)-rainbow dominating function; mixed \(k\)-rainbow domination number PDFBibTeX XMLCite \textit{J. Amjadi} et al., Util. Math. 108, 53--71 (2018; Zbl 1464.05278)