Characterization of n-Vertex Graphs with Metric Dimension n-3

TL;DR

This paper characterizes all connected graphs of order n that have a metric dimension of n-3, providing a complete classification based on the properties of resolving sets and vertex representations.

Contribution

It offers a complete characterization of graphs with metric dimension n-3, advancing understanding of graph resolving sets and metric dimensions.

Findings

Characterization of all graphs with metric dimension n-3

Identification of structural properties related to resolving sets

Complete classification for graphs of order n with this metric dimension

Abstract

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we characterize all graphs of order $n$ with metric dimension $n-3$.