For a positive integer $r$, a distance-$r$ independent set in an undirected graph $G$ is a set $I\subseteq V\left(G\right)$ of vertices pairwise at distance greater than $r$, while a distance-$r$ dominating set is a set $D\subseteq V\left(G\right)$ such that every vertex of the graph is within distance at most $r$ from a vertex from $D$. We study the duality between the maximum size of a distance-$2r$ independent set and the minimum size of a distance-$r$ dominating set in nowhere dense graph classes, as well as the kernelization complexity of the distance-$r$ independent set problem on these graph classes. Specifically, we prove that the distance-$r$ independent set problem admits an almost linear kernel on every nowhere dense graph class.

对于正整数 $\mathrm{[R}$，距离-$\mathrm{[R}$ 无向图中的独立集 $G$ 是一套 $\mathrm{一世}\subseteq V（G）$ 成对的顶点距离大于 $\mathrm{[R}$，而距离$\mathrm{[R}$ 主宰集是一个集 $d\subseteq V（G）$ 这样图形的每个顶点最多都在距离之内 $\mathrm{[R}$ 来自一个顶点 $d$。我们研究了距离的最大大小之间的对偶关系$2\mathrm{[R}$ 独立设置和最小距离$\mathrm{[R}$ 无处密集图类中的控制集，以及距离的核化复杂度$\mathrm{[R}$这些图类的独立集问题。具体来说，我们证明距离-$\mathrm{[R}$ 独立集问题允许在每个无处密集图类上使用几乎线性的核。