Locating-dominating codes in cycles

The smallest cardinality of an r-locating-dominating code in a cycle C-n of length n is denoted by M-r(LD)(C-n). In this paper, we prove that for any r >= 5 and n >= n(r) when n(r) is large enough (n(r) = O(r(3))) we have n/3 <= M-r(LD)(C-n) <= n/3 + 1 if n equivalent to 3 (mod 6) and M-r(LD)(C-n) = inverted right perpendicularn/3inverted left perpendicular otherwise. Moreover, we determine the exact values of M-3(LD)(C-n) and M-4(LD)(C-n) for all n.