# Optimal bounds on codes for location in circulant graphs

## 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.

## References

- 1.Ben-Haim, Y., Litsyn, S.: Exact minimum density of codes identifying vertices in the square grid. SIAM J. Discret. Math.
**19**(1), 69–82 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Bertrand, N., Charon, I., Hudry, O., Lobstein, A.: Identifying and locating-dominating codes on chains and cycles. Eur. J. Comb.
**25**(7), 969–987 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Charon, I., Hudry, O., Lobstein, A.: Identifying codes with small radius in some infinite regular graphs. Electron. J. Combin.
**9**(1), Research Paper 11 (2002)MathSciNetzbMATHGoogle Scholar - 4.Chen, C., Lu, C., Miao, Z.: Identifying codes and locating-dominating sets on paths and cycles. Discret. Appl. Math.
**159**(15), 1540–1547 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Cohen, G., Gravier, S., Honkala, I., Lobstein, A., Mollard, M., Payan, C., Zémor, G.: Improved identifying codes for the grid. Electron. J. Combin.
**6**, Research Paper 19, Comment (1999)MathSciNetGoogle Scholar - 6.Cohen, G., Honkala, I., Lobstein, A., Zémor, G.: On codes identifying vertices in the two-dimensional square lattice with diagonals. IEEE Trans. Comput.
**50**(2), 174–176 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Exoo, G., Junnila, V., Laihonen, T.: Locating-dominating codes in cycles. Australas. J. Combin.
**49**, 177–194 (2011)MathSciNetzbMATHGoogle Scholar - 8.Ghebleh, M., Niepel, L.: Locating and identifying codes in circulant networks. Discret. Appl. Math.
**161**(13-14), 2001–2007 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Gravier, S., Moncel, J., Semri, A.: Identifying codes of cycles. Eur. J. Comb.
**27**(5), 767–776 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Honkala, I.: An optimal locating-dominating set in the infinite triangular grid. Discret. Math.
**306**(21), 2670–2681 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Honkala, I., Laihonen, T.: On locating-dominating sets in infinite grids. Eur. J. Comb.
**27**(2), 218–227 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Honkala, I., Laihonen, T.: On a new class of identifying codes in graphs. Inform. Process. Lett.
**102**(2-3), 92–98 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Junnila, V., Laihonen, T.: Optimal identifying codes in cycles and paths. Graphs Combin.
**28**(4), 469–481 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Junnila, V., Laihonen, T.: Collection of codes for tolerant location. In: Proceedings of the Bordeaux Graph Workshop, pp. 176–179 (2016)Google Scholar
- 15.Junnila, V., Laihonen, T.: Tolerant location detection in sensor networks. Submitted (2016)Google Scholar
- 16.Junnila, V., Laihonen, T., Paris, G.: Solving two conjectures regarding codes for location in circulant graphs. Submitted (2017)Google Scholar
- 17.Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inform. Theory
**44**(2), 599–611 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Manuel, P.: Locating and liar domination of circulant networks. Ars Combin.
**101**, 309–320 (2011)MathSciNetzbMATHGoogle Scholar - 19.Rall, D.F., Slater, P.J.: On location-domination numbers for certain classes of graphs. Congr. Numer.
**45**, 97–106 (1984)MathSciNetzbMATHGoogle Scholar - 20.Roberts, D.L., Roberts, F.S.: Locating sensors in paths and cycles: The case of 2-identifying codes. Eur. J. Comb.
**29**(1), 72–82 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Slater, P.J.: Domination and location in acyclic graphs. Networks
**17**(1), 55–64 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Slater, P.J.: Dominating and reference sets in a graph. J. Math. Phys. Sci.
**22**, 445–455 (1988)MathSciNetzbMATHGoogle Scholar - 23.Slater, P.J.: Fault-tolerant locating-dominating sets. Discret. Math.
**249**(1–3), 179–189 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Xu, M., Thulasiraman, K., Hu, X.-D.: Identifying codes of cycles with odd orders. Eur. J. Comb.
**29**(7), 1717–1720 (2008)MathSciNetCrossRefzbMATHGoogle Scholar