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Optimal bounds on codes for location in circulant graphs

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Abstract

Identifying and locating-dominating codes have been studied widely in circulant graphs of type \(C_{n}(1,2,3,\dots , r)\) over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs \(C_{n}(1,d)\) for \(d = 3\) and proposed as an open question the case of \(d > 3\). In this paper we study identifying, locating-dominating and self-identifying codes in the graphs \(C_{n}(1,d)\), \(C_{n}(1,d-1,d)\) and \(C_{n}(1,d-1,d,d + 1)\). We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in \(C_{n}(1,3)\) and \(C_{n}(1,4)\).

Keywords

Identifying code Locating-dominating code Circulant graph Square grid Triangular grid King grid 

Mathematics Subject Classification (2010)

94B25 94B65 05C69 05B40 

Notes

Acknowledgements

We would like to thank the referees for their suggestions which improved the presentation of the paper.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.LIRISUniversity of LyonLyonFrance

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