# 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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