Nonlocal metric dimension of graphs

TL;DR

This paper introduces the nonlocal metric dimension of graphs, characterizes specific cases, computes it for certain graph classes, and provides bounds and embeddings related to this new graph invariant.

Contribution

It defines the nonlocal metric dimension, characterizes graphs with extreme values, and computes it for block graphs, corona products, and wheels, also providing bounds and embeddings.

Findings

Characterized graphs with nonlocal metric dimension 1 and n(G)-2

Determined nonlocal metric dimension for block graphs, corona products, and wheels

Established upper bounds and embedding methods for graphs

Abstract

Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with ${\rm dim}_{\rm n\ell}(G) = 1$ or with ${\rm dim}_{\rm n\ell}(G) = n(G)-2$ are characterized. The nonlocal metric dimension is determined for block graphs, for corona products, and for wheels. Two upper bounds on the nonlocal metric dimension are proved. An embedding of an arbitrary graph into a supergraph with a small nonlocal metric dimension and small diameter is presented.