Identifying codes of cycles with odd orders

The problem of the r -identifying code of a cycle C n has been solved totally when n is even. Recently, S. Gravier et al. give the r -identifying code for the cycle C n with the minimum cardinality for odd n , when n ≥ 3 r + 2 and gcd ( 2 r + 1 , n ) ≠ 1 . In this paper, we deal with the r -identifying code of the cycle C n for odd n , when n ≥ 3 r + 2 and gcd ( 2 r + 1 , n ) = 1 .