A set of subgraphs C 1, C 2,…, C k in a graph G is said to identify the vertices (resp. the edges) if the sets {j : v∈C j} (resp. {j : e∈C j} ) are nonempty for all the vertices v (edges e) and no two are the same set. We consider the problem of minimizing k when the subgraphs C i are required to be cycles or closed walks. The motivation comes from maintaining multiprocessor systems, and we study the cases when G is the binary hypercube, or the two-dimensional p-ary space endowed with the Lee metric.