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Codes for Identification in the King Lattice

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Abstract.

Let G=(V, E) be an undirected graph and C a subset of vertices. If the sets B r (F)∩C are distinct for all subsets F⊆V with at most k elements, then C is called an (r,≤k)-identifying code in G. Here B r (F) denotes the set of all vertices that are within distance r from at least one vertex in F. We consider the two-dimensional square lattice with diagonals (the king lattice). We show that the optimal density of an (r,≤2)-identifying code is 1/4 for all r≥3. For r=1,2, we give constructions with densities 3/7 and 3/10, and we prove the lower bounds 9/22 and 31/120, respectively.

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Authors and Affiliations

  1. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Iiro Honkala & Tero Laihonen

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  1. Iiro Honkala
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  2. Tero Laihonen
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Correspondence to Tero Laihonen.

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The research of the authors was supported by the Academy of Finland under grants 44002 and 46186

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Honkala, I., Laihonen, T. Codes for Identification in the King Lattice. Graphs and Combinatorics 19, 505–516 (2003). https://doi.org/10.1007/s00373-003-0531-2

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  • DOI: https://doi.org/10.1007/s00373-003-0531-2

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