Minimum-Density Identifying Codes in Square Grids

  • Marwane Bouznif
  • Frédéric Havet
  • Myriam Preissmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9778)


An identifying code in a graph G is a subset of vertices with the property that for each vertex \(v \in V(G)\), the collection of elements of C at distance at most 1 from v is non-empty and distinct from the collection of any other vertex. We consider the minimum density \(d^*(\mathcal{S}_k)\) of an identifying code in the square grid \(\mathcal{S}_k\) of height k (i.e. with vertex set \( \mathbb {Z} \times \{1, \dots , k\}\)). Using the Discharging Method, we prove \(\displaystyle \frac{7}{20} + \frac{1}{20k} \le d^*(\mathcal{S}_k) \le \min \left\{ \frac{2}{5}, \frac{7}{20} + \frac{3}{10k} \right\} \), and \(\displaystyle d^*(\mathcal{S}_3) =\frac{7}{18}\).


Identifying code Square grid Discharging method 


  1. 1.
    Ben-Haim, Y., Litsyn, S.: Exact minimum density of codes identifying vertices in the square grid. SIAM J. Discrete Math. 19, 69–82 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Charon, I., Honkala, I., Hudry, O., Lobstein, A.: General bounds for identifying codes in some infinite regular graphs. Electron. J. Comb. 8(1), Research Paper 39, 21 pp. (2001)Google Scholar
  3. 3.
    Cohen, G., Gravier, S., Honkala, I., Lobstein, A., Mollard, M., Payan, C., Zémor, G.: Improved identifying codes for the grid, Comment to [4]Google Scholar
  4. 4.
    Cohen, G., Honkala, I., Lobstein, A., Zémor, G.: New bounds for codes identifying vertices in graphs. Electron. J. Comb. 6(1), R19 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cukierman, A., Yu, G.: New bounds on the minimum density of an identifying code for the infinite hexagonal grid discrete. Appl. Math. 161, 2910–2924 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Daniel, M., Gravier, S., Moncel, J.: Identifying codes in some subgraphs of the square lattice. Theoret. Comput. Sci. 319, 411–421 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Honkala, I.: A family of optimal identifying codes in \(\mathbb{Z}^2\). J. Comb. Theor. Ser. A 113(8), 1760–1763 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Junnila, V.: New lower bound for 2-identifying code in the square grid. Discrete Appl. Math. 161(13–14), 2042–2051 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Junnila, V., Laihonen, T.: Optimal lower bound for 2-identifying codes in the hexagonal grid. Electron. J. Comb. 19(2), Paper 38 (2012)Google Scholar
  10. 10.
    Karpovsky, M., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inf. Theory 44, 599–611 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ray, S., Starobinski, D., Trachtenberg, A., Ungrangsi, R.: Robust location detection with sensor networks. IEEE J. Sel. Areas Commun. 22(6), 1016–1025 (2004)CrossRefGoogle Scholar
  12. 12.
    Slater, P.J.: Fault-tolerant locating-dominating sets. Discrete Math. 249(1–3), 179–189 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Marwane Bouznif
    • 1
  • Frédéric Havet
    • 2
  • Myriam Preissmann
    • 3
  1. 1.A-SISSaint ÉtienneFrance
  2. 2.Projet COATI, I3S (CNRS, UNSA) and INRIASophia AntipolisFrance
  3. 3.Univ. Grenoble Alpes and CNRS, G-SCOPGrenobleFrance

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