On $k$-rainbow domination in middle graphs

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f:V(G)\to \mathcal{P}(\{1,2,\dots ,k\})$ is a $k\mathit{\text{-rainbow}}\mathit{\text{dominating}}\mathit{\text{function}}$ on $G$ if for each vertex $v\in V(G)$ for which $f(v)=\mathrm{\varnothing }$, it holds that $\bigcup_{u \in N(v)}f(u) = {1, 2, \dotsc, k}$\mathrm{.}Theweightofa$k$-rainbowdominatingfunctionisthevalue$\sum_{v \in V(G)}|f(v)|. The \textit{k-rainbow domination number}\gamma_{rk}(G)$istheminimumweightofa$k$-rainbowdominatingfunctionon$G$\mathrm{.}Inthispaper,weinitiatethestudyof$k$-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.