Nonlocal metric dimension of graphs

Abstract

Nonlocal metric dimension ${\dim }_{\mathrm{n}\mathrm{\ell }}(G)$ 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 ${\dim }_{\mathrm{n}\mathrm{\ell }}(G)=1$ or with ${\dim }_{\mathrm{n}\mathrm{\ell }}(G)=n(G)-2$ 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.