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

  • Authors:

  • David L. Roberts;Fred S. Roberts

  • Affiliations:

  • College of Computing, Georgia Institute of Technology, Atlanta, GA 30332, USA;DIMACS, Rutgers University, Piscataway, NJ 08854, USA

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2008

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Abstract

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)0@?, and for every pair of vertices x and y, xy@?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.