The connected metric dimension at a vertex of a graph

Abstract

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 $cdim_G(v)$, \emph{the connected metric dimension of $G$ at a vertex $v$}, which is defined as follows: a set of vertices $S$ of $G$ is a \emph{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 call a resolving set $S$ \emph{connected} if $S$ induces a connected subgraph of $G$. Then, $cdim_G(v)$ is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex $v$. The \emph{connected metric dimension of $G$}, denoted by $cdim(G)$, is $\min\{cdim_G(v): v \in V(G)\}$. Noting that $1 \le dim(G) \le cdim(G) \le cdim_G(v) \le |V(G)|-1$ for any vertex $v$ of $G$, we show the existence of a pair $(G,v)$ such that $cdim_G(v)$ 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 $cdim_G(v) \in \{1, |V(G)|-1\}$. We show that $cdim(G)=2$ 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 $cdim(G)=dim(G)$. We show that $cdim(G)-dim(G)$ can be arbitrarily large. We determine $cdim(G)$ and $cdim_G(v)$ 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.