On $k$-limited domination: complexity and Cartesian products

A dominating set is called $k$-limited if every vertex in the set has at most $k$ neighbors outside it. The minimum cardinality among all $k$-limited dominating sets of a graph $G$ is the $k$-limited domination number, denoted by $γ_k^{\mathrm L}(G)$. We prove that, for every fixed integer $k\ge 2$,