As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph $G=(V,E)$, a partition Π of V is said to be a k-partition generator of G if any pair of different vertices $u,v\in V$ is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets $S_1,\dots ,S_k\in \mathrm{\Pi }$ such that $d(u,S_i)\ne d(v,S_i)$ for every $i\in \{1,\dots ,k\}$. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for $k\in \{1,\dots ,r\}$.

作为图的分区维的概念的概括，本文介绍了k分区维的概念。给定一个非平凡的连通图$G=（V，Ë）$，如果任意一对不同的顶点，则V的一个分区Π被称为G的一个k分区生成器$ü，v\in V$用至少k个顶点集合Π来区分，即存在至少k个顶点集合${\mathrm{小号}}_{\mathrm{1个}}，\dots ，{\mathrm{小号}}_{ķ}\in \mathrm{\Pi }$ 这样 $d（ü，{\mathrm{小号}}_{\mathrm{一世}}）\ne d（v，{\mathrm{小号}}_{\mathrm{一世}}）$ 每一个 $\mathrm{一世}\in \{\mathrm{1个}，\dots ，ķ\}$。甲ķ -partition的发生器ģ与所有其中最小基数ķ -partition发电机被称为ķ的-partition基础ģ及其基数的ķ的-partition尺寸ģ。如果k是最大整数，则非平凡的连通图G是k分区维，因此G具有k分区基础。我们给出了图为r分区维的充要条件，并获得了k分区维的几个结果$ķ\in \{\mathrm{1个}，\dots ，\mathrm{[R}\}$。