Given a graph $G$, the $k$-hop dominating set problem asks for a vertex subset $D_k$ such that every vertex of $G$ is in distance at most $k$ to some vertex in $D_k$ ($k\in \mathbb{N}$). For $k=1$, this corresponds to the classical dominating set problem in graphs. We study the $k$-hop dominating set problem in grid graphs (motivated by generalized guard sets in polyominoes). We show that the VC dimension of this problem is 3 in grid graphs without holes, and 4 in general grid graphs. Furthermore, we provide a reduction from planar monotone 3SAT, thereby showing that the problem is NP-complete even in thin grid graphs (i.e., grid graphs that do not a contain an induced $C_4$). Complementary, we present a linear-time $4$-approximation algorithm for $2$-thin grid graphs (which do not contain a $3\times 3$-grid subgraph) for all $k\in \mathbb{N}$.

翻译：给定图$G$，$k$-跳支配集问题要求找到一个顶点子集$D_k$，使得$G$中每个顶点到$D_k$中某个顶点的距离至多为$k$（$k\in \mathbb{N}$）。当$k=1$时，该问题对应于经典图支配集问题。我们研究了网格图（由多联骨牌中的广义守卫集启发）上的$k$-跳支配集问题。我们证明了该问题在无孔网格图上的VC维数为3，在一般网格图上为4。此外，我们给出了从平面单调3SAT问题的归约，从而证明即使在薄网格图（即不包含诱导$C_4$的网格图）上该问题也是NP完全的。作为补充，我们针对所有$k\in \mathbb{N}$，在$2$-薄网格图（即不包含$3\times 3$网格子图的网格图）上提出了一个线性时间的4-近似算法。