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Strong Geodetic Problem in Grid-Like Architectures

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Abstract

A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then \(\mathrm{sg}(G)\) is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair \(\{x,y\}\subseteq S\) so that these \(\left( {\begin{array}{c}|S|\\ 2\end{array}}\right) \) geodesics cover all the vertices of G. In this paper, the strong geodetic problem is studied on Cartesian product graphs. A general upper bound is proved on the Cartesian product of a path with an arbitrary graph and showed that the bound is tight on thin grids and thin cylinders.

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Acknowledgements

S.K. acknowledges the financial support from the Slovenian Research Agency (Research Core Funding No. P1-0297).

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sandi Klavžar

  2. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Sandi Klavžar

  3. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Sandi Klavžar

  4. Department of Information Science, College of Computing Science and Engineering, Kuwait University, Safat, Kuwait

    Paul Manuel

Authors
  1. Sandi Klavžar
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  2. Paul Manuel
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Correspondence to Sandi Klavžar.

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Communicated by Sanming Zhou.

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Klavžar, S., Manuel, P. Strong Geodetic Problem in Grid-Like Architectures. Bull. Malays. Math. Sci. Soc. 41, 1671–1680 (2018). https://doi.org/10.1007/s40840-018-0609-x

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  • DOI: https://doi.org/10.1007/s40840-018-0609-x

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