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On Codes Identifying Vertices in the Two-Dimensional Square Lattice with Diagonals
February 2001 (vol. 50 no. 2)
pp. 174-176

Abstract—Fault diagnosis of multiprocessor systems motivates the following graph-theoretic definition. A subset $C$ of points in an undirected graph $G=(V,E)$ is called an identifying code if the sets $B(v) \cap C$ consisting of all elements of $C$ within distance one from the vertex $v$ are different. We also require that the sets $B(v) \cap C$ are all nonempty. We take $G$ to be the infinite square lattice with diagonals and show that the density of the smallest identifying code is at least 2/9 and at most 4/17.

[1] U. Blass, I. Honkala, and S. Litsyn, “Bounds on Identifying Codes,” Discrete Math., to appear.
[2] U. Blass, I. Honkala, and S. Litsyn, “On Binary Codes for Identification,” J. Combinatorial Designs, vol. 8, pp. 151-156, 2000.
[3] G. Cohen, S. Gravier, I. Honkala, A. Lobstein, M. Mollard, C. Payan, G. Zémor, “Improved Identifying Codes for the Grid,” Electronic J. Combinatorics, comments to vol. 6, no. 1, R19, 1999.
[4] G. Cohen, I. Honkala, A. Lobstein, and G. Zémor, “New Bounds for Codes Identifying Vertices in Graphs,” Electronic J. Combinatorics, vol. 6, no. 1,R19, 1999.
[5] G. Cohen, I. Honkala, A. Lobstein, G. Zémor, “On Identifying Codes,” Proc. DIMACS Workshop codes and Association Schemes, to appear.
[6] G. Cohen, I. Honkala, A. Lobstein, G. Zémor, “Bounds for Codes Identifying Vertices in the Hexagonal Grid,” SIAM J. Discrete Math., vol. 13, no. 4, pp. 492-504, 2000.
[7] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs. Marcel Dekker, 1998.
[8] M.G. Karpovksy, K. Chakrabarty, and L.B. Levitin, “A New Class of Codes for Covering Vertices in Graphs,” IEEE Trans. Information Theory, vol. 44, pp. 599-611, Mar. 1998.
[9] M.G. Karpovsky, K. Chakrabarty, L.B. Levitin, and D.R. Avresky, “On the Covering of Vertices for Fault Diagnosis in Hypercubes,” Information Processing Letters, vol. 69, pp. 99-103, 1999.
[10] P.J. Slater, “Locating Dominating Sets and Locating-Dominating Sets,” Graph Theory, Combinatorics, and Applications, Proc. Seventh Quadrennial Int'l Conf. Theory and Applications of Graphs, Y. Alavi and A. Schwenk, eds., vol. 2, pp. 1,073-1,079, 1995.

Index Terms:
Graph, square lattice, code, identifying code.
Citation:
Gérard D. Cohen, Iiro Honkala, Antoine Lobstein, Gilles Zémor, "On Codes Identifying Vertices in the Two-Dimensional Square Lattice with Diagonals," IEEE Transactions on Computers, vol. 50, no. 2, pp. 174-176, Feb. 2001, doi:10.1109/12.908992
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