Abstract
A subset S of vertices in a graph G is called a resolving set for G if for arbitrary two distinct vertices \(u, v\in V\), there exists a vertex x from S such that the distances \(d(u, x)\ne d(v, x)\). The metric dimension of G is the minimum cardinality of a resolving set of G. A minimal resolving set is a resolving set which has no proper subsets that are resolving sets. Let \(\Box _{n}\) denote the folded n-cube. In this paper, we consider the metric dimension of \(\Box _{n}\). By constructing explicitly minimal resolving sets for \(\Box _{n}\), we obtain upper bounds on the metric dimension of this graph.
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Acknowledgements
The authors would like to thank the referees for a careful reading of the paper and for many constructive comments. This research is supported by the NSF of China (Nos. 11471097 and 11971146), the NSF of Hebei Province (Nos. A2017403010 and A2019205089) and Overseas Expertise Introduction Program of Hebei Auspices (25305008).
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Zhang, Y., Hou, L., Hou, B. et al. On the metric dimension of the folded n-cube. Optim Lett 14, 249–257 (2020). https://doi.org/10.1007/s11590-019-01476-z
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DOI: https://doi.org/10.1007/s11590-019-01476-z