The notion of metric dimension, $\dim (G)$, of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing ${\text{cdim}}_G(v)$, the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex $z\in S$ such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, ${\text{cdim}}_G(v)$ is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by $\text{cdim}(G)$, is $\min \{{\text{cdim}}_G(v):v\in V(G)\}$. Noting that $1\le \dim (G)\le \text{cdim}(G)\le {\text{cdim}}_G(v)\le |V(G)|-1$ for any vertex v of G, we show the existence of a pair $(G,v)$ such that ${\text{cdim}}_G(v)$ takes all positive integer values from $\dim (G)$ to $|V(G)|-1$, as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying ${\text{cdim}}_G(v)\in \{1,|V(G)|-1\}$. We show that $\text{cdim}(G)=2$ implies G is planar, whereas it is well known that there is a non-planar graph H with $\dim (H)=2$. We also characterize trees and unicyclic graphs G satisfying $\text{cdim}(G)=\dim (G)$. We show that $\text{cdim}(G)-\dim (G)$ can be arbitrarily large. We determine $\text{cdim}(G)$ and ${\text{cdim}}_G(v)$ for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems.

度量维度的概念 $\mathrm{暗淡}（G）$现在对图G的，以及许多变体进行了很好的研究。在本文中，我们通过介绍以下内容开始对该概念的本地分析${\text{光盘}}_G（v）$，在一个顶点v G的连接的度量尺寸，其被定义如下：一组顶点š的ģ是解决集如果，对任何一对不同顶点的X和ÿ的ģ，有一个顶点$ž\in \mathrm{小号}$使得之间的距离Ž和X是从之间的距离不同Ž和ÿ在ģ。我们说，如果S诱导了G的连通子图，则解析集S被连通。然后，${\text{光盘}}_G（v）$定义为包含顶点v的所有连接的解析集的基数的最小值。G的连接度量维，表示为$\text{光盘}（G）$，是 $\mathrm{分}\{{\text{光盘}}_G（v）：v\in V（G）\}$。注意$\mathrm{1个}\le \mathrm{暗淡}（G）\le \text{光盘}（G）\le {\text{光盘}}_G（v）\le |V（G）|-\mathrm{1个}$对于任何顶点v的ģ，我们示出了对存在$（G，v）$ 这样 ${\text{光盘}}_G（v）$ 从中获取所有正整数值 $\mathrm{暗淡}（G）$ 至 $|V（G）|-\mathrm{1个}$，因为v在固定图G中变化。我们表征图G及其顶点v满足${\text{光盘}}_G（v）\in \{\mathrm{1个}，|V（G）|-\mathrm{1个}\}$。我们证明$\text{光盘}（G）=2$意味着ģ是平面的，而它是公知的，有一个非平面图形ħ与$\mathrm{暗淡}（H）=2$。我们还描述了满足以下条件的树和单圈图G$\text{光盘}（G）=\mathrm{暗淡}（G）$。我们证明$\text{光盘}（G）-\mathrm{暗淡}（G）$可以任意大。我们确定$\text{光盘}（G）$ 和 ${\text{光盘}}_G（v）$对于某些类的图。我们进一步检查顶点或边删除对连接的度量维度的影响。我们以一些未解决的问题作为结论。