Locating sensors in paths and cycles: The case of 2-identifying codes

For a graph G and a set D ⊆ V ( G ) , define N r [ x ] = { x i ∈ V ( G ) : d ( x , x i ) ≤ r } (where d ( x , y ) is graph theoretic distance) and D r ( x ) = N r [ x ] ∩ D . D is known as an r -identifying code if for every vertex x , D r ( x ) ≠ 0̸ , and for every pair of vertices x and y , x ≠ y ⇒ D r ( x ) ≠ D r ( y ) . The various applications of these codes include attack sensor placement in networks and fault detection/localization in multiprocessor or distributed systems. Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating–dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969–987] and Gravier et al. [S. Gravier, J. Moncel, A. Semri, Identifying codes of cycles, European Journal of Combinatorics 27 (2006) 767–776] provide partial results about the minimum size of D for r -identifying codes for paths and cycles and present complete closed form solutions for the case r = 1 , based in part on Daniel [M. Daniel, Codes identifiants, Rapport pour le DEA ROCO, Grenoble, June 2003]. We provide complete solutions for the case r = 2 .