Let $G=(V,E)$ be a simple, unweighted, connected graph. Let $d(u,v)$ denote the distance between vertices $u,v$. A resolving set of $G$ is a subset $S$ of $V$ such that knowing the distance from a vertex $v$ to every vertex in $S$ uniquely identifies $v$. The metric dimension of $G$ is defined as the size of the smallest resolving set of $G$. We define the $k$-truncated resolving set and $k$-truncated metric dimension of a graph similarly, but with the notion of distance replaced with $d_k(u,v) := \min(d(u,v),k+1)$. In this paper, we demonstrate that computing $k$-truncated dimension of trees is NP-Hard for general $k$. We then present a polynomial-time algorithm to compute $k$-truncated dimension of trees when $k$ is a fixed constant.

翻译：设 $G=(V,E)$ 为一个简单、无权、连通的图。令 $d(u,v)$ 表示顶点 $u$ 与 $v$ 之间的距离。图 $G$ 的一个分辨集是 $V$ 的子集 $S$，使得从任意顶点 $v$ 到 $S$ 中每个顶点的距离能够唯一确定 $v$。图 $G$ 的度量维数定义为最小分辨集的大小。类似地，我们定义图的 $k$-截断分辨集和 $k$-截断度量维数，但将距离概念替换为 $d_k(u,v) := \min(d(u,v),k+1)$。本文证明，对于一般 $k$ 值，计算树的 $k$-截断维数是一个NP难问题。随后，我们给出一个多项式时间算法，用于在 $k$ 为固定常数时计算树的 $k$-截断维数。