Edge and mixed metric dimension of Johnson graphs

TL;DR

This paper investigates the edge and mixed metric dimensions of Johnson graphs, establishing a new lower bound and deriving exact values for specific cases, thereby advancing understanding of graph metric properties.

Contribution

Introduces a new tight lower bound for the edge metric dimension of Johnson graphs and determines exact values for the case when k=2.

Findings

Established a new lower bound for $eta_E(J_{n,k})$

Derived exact values for $eta_E(J_{n,2})$ and $eta_M(J_{n,2})$

Proved $eta_E(J_{n,2}) = eta_M(J_{n,2})$

Abstract

In this paper, both edge and mixed metric dimensions of Johnson graphs $J_{n,k}$ are considered. A new tight lower bound for $\beta_E(J_{n,k})$ based on hitting sets has been obtained. Using this bound, exact values for $\beta_E(J_{n,2})$ and $\beta_M(J_{n,2})$ have been derived, and it is proved that $\beta_E(J_{n,2}) = \beta_M(J_{n,2})$.