Truncated Metric Dimension for Finite Graphs

TL;DR

This paper introduces and studies the truncated metric dimension of finite graphs, analyzing how it varies with the truncation parameter and characterizing it for specific graph classes like paths, cycles, and trees.

Contribution

It defines the truncated metric dimension using a thresholded distance metric and investigates its properties, including exact values for certain graph families and extremal cases.

Findings

Characterized truncated metric dimension for paths and cycles.

Analyzed how truncated metric dimension varies with the threshold and graph diameter.

Explored extremal graphs with maximum and minimum truncated metric dimension.

Abstract

A graph $G=(V,E)$ with geodesic distance $d(\cdot,\cdot)$ is said to be resolved by a non-empty subset $R$ of its vertices when, for all vertices $u$ and $v$, if $d(u,r)=d(v,r)$ for each $r\in R$, then $u=v$. The metric dimension of $G$ is the cardinality of its smallest resolving set. In this manuscript, we present and investigate the notions of resolvability and metric dimension when the geodesic distance is truncated with a certain threshold $k$; namely, we measure distances in $G$ using the metric $d_k(u,v):=\min\{d(u,v),k+1\}$. We denote the metric dimension of $G$ with respect to $d_k$ as $\beta_k(G)$. We study the behavior of this quantity with respect to $k$ as well as the diameter of $G$. We also characterize the truncated metric dimension of paths and cycles as well as graphs with extreme metric dimension, including graphs of order $n$ such that $\beta_k(G)=n-2$ and $\beta_k(G)=n-1$. We conclude with a study of various problems related to the truncated metric dimension of trees.