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 $C_{n}(1,4)$ .
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