The edge metric dimension of the generalized Petersen graph $P(n,3)$ is 4

TL;DR

This paper proves that the edge metric dimension of the generalized Petersen graph P(n,3) is 4 for all n ≥ 11, by combining combinatorial nonexistence proofs with explicit constructions.

Contribution

It establishes the exact edge metric dimension of P(n,3) for large n, filling a gap in the understanding of Petersen graph variants.

Findings

Edge dimension of P(n,3) is 4 for n ≥ 11

Constructed explicit edge resolving sets of size 4

Proved nonexistence of smaller resolving sets for n ≥ 11

Abstract

It is known that the problem of computing the edge dimension of a graph is NP-hard, and that the edge dimension of any generalized Petersen graph $P(n,k)$ is at least 3. We prove that the graph $P(n,3)$ has edge dimension 4 for $n\ge 11$, by showing semi-combinatorially the nonexistence of an edge resolving set of order 3 and by constructing explicitly an edge resolving set of order 4.