# On the edge metric dimension of convex polytopes and its related graphs

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## Abstract

Let \(G=(V, E)\) be a connected graph. The distance between the edge \(e=uv\in E\) and the vertex \(x\in V\) is given by \(d(e, x) = \min \{d(u, x), d(v, x)\}\). A subset \(S_{E}\) of vertices is called an edge metric generator for *G* if for every two distinct edges \(e_{1}, e_{2}\in E\), there exists a vertex \(x\in S_{E}\) such that \(d(e_{1}, x)\ne d(e_{2}, x)\). An edge metric generator containing a minimum number of vertices is called an edge metric basis for *G* and the cardinality of an edge metric basis is called the edge metric dimension denoted by \(\mu _{E}(G)\). In this paper, we study the edge metric dimension of some classes of plane graphs. It is shown that the edge metric dimension of convex polytope antiprism \(A_{n}\), the web graph \({\mathbb {W}}_{n}\), and convex polytope \({\mathbb {D}}_{n}\) are bounded, while the prism related graph \(D^{*}_{n}\) has unbounded edge metric dimension.

## Keywords

Metric dimension Edge metric dimension Edge metric generator Convex polytopes## Notes

### Acknowledgements

The authors would like to thank the reviewers for a careful reading of the paper and for many constructive comments. This research is supported by the NSF of China (No.11471097 and No.11971146), the NSF of Hebei Province (No.A2017403010).

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