On a new class of identifying codes in graphs

Abstract

Assume that G = ( V , E ) is an undirected graph, and C ⊆ V . For every v ∈ V , we denote I r ( v ) = { u ∈ C : d ( u , v ) ⩽ r } , where d ( u , v ) denotes the number of edges on any shortest path from u to v. For every F ⊆ V , we denote I r ( F ) = ⋃ v ∈ F I r ( v ) . We study codes C with the property that if I r ( F ) = I r ( F ′ ) and F ≠ F ′ , then both F and F ′ have size at least l + 1 . Such codes can be used in the maintenance of multiprocessor architectures. We consider the cases when G is the infinite square or king grid, infinite triangular lattice or hexagonal mesh, or a binary hypercube.