The connected metric dimension at a vertex of a graph

TL;DR

This paper introduces the connected metric dimension at a vertex in a graph, analyzing its properties, relationships with other graph invariants, and effects of modifications, providing new insights into local graph resolving sets.

Contribution

It defines and studies the connected metric dimension at a vertex, characterizing graphs with extreme values and exploring its relation to planarity, trees, and graph modifications.

Findings

Connected metric dimension can vary from the metric dimension to the number of vertices minus one.

Graphs with cdim(G)=2 are planar, but some non-planar graphs also have metric dimension 2.

The difference between cdim(G) and dim(G) can be arbitrarily large.

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.