On $k$-limited domination: complexity and Sierpiński graphs

A dominating set $D$ of a graph $G$ is called $k$-limited if every vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality among all $k$-limited dominating sets of $G$ is the $k$-limited domination number, denoted by ${\gamma }_k^L(G)$. In this paper, we prove that the $1$-Limited Dominating Set problem is $\mathsf{N}\mathsf{P}$-complete, answering an open question posed in the literature. We further show that, for every positive integer $k$, the $k$-Limited Dominating Set problem is $\mathsf{N}\mathsf{P}$-complete even when restricted to planar graphs. In addition, we study the $k$-limited domination number of Sierpiński graphs. We determine the exact value of ${\gamma }_1^L(S(n,m))$ for all integers $n\ge 1$ and $m\ge 2$, and obtain the $k$-limited domination number for the planar family $S(n,3)$.