## 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.

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## 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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Zhang, Y., Gao, S. On the edge metric dimension of convex polytopes and its related graphs.
*J Comb Optim* **39, **334–350 (2020) doi:10.1007/s10878-019-00472-4

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### Keywords

- Metric dimension
- Edge metric dimension
- Edge metric generator
- Convex polytopes