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Symmetry properties of resolving sets and metric bases in hypercubes

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Abstract

In this paper we consider some special characteristics of distances between vertices in the \(n\)-dimensional hypercube graph \(Q_n\) and, as a consequence, the corresponding symmetry properties of its resolving sets. It is illustrated how these properties can be implemented within a simple greedy heuristic in order to find efficiently an upper bound of the so called metric dimension \(\beta (Q_n)\) of \(Q_n\), i.e. the minimal cardinality of a resolving set in \(Q_n\). This heuristic was applied to generate upper bounds of \(\beta (Q_n)\) for \(n\) up to \(22\), which are for \(n\ge 19\) better than the existing ones. Starting from these new bounds, some existing upper bounds for \(23\le n\le 90\) are improved by a dynamic programming procedure.

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Acknowledgments

This work is partially supported by the Serbian Ministry of education, science and technological development, Projects No. 174010, and 174033.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, 11000, Belgrade, Serbia

    Nebojša Nikolić, Mirjana Čangalović & Igor Grujičić

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  1. Nebojša Nikolić
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  2. Mirjana Čangalović
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  3. Igor Grujičić
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Correspondence to Nebojša Nikolić.

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Nikolić, N., Čangalović, M. & Grujičić, I. Symmetry properties of resolving sets and metric bases in hypercubes. Optim Lett 11, 1057–1067 (2017). https://doi.org/10.1007/s11590-014-0790-2

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