Bounds for Codes Identifying Vertices in the Hexagonal Grid
SIAM Journal on Discrete Mathematics
Exact Minimum Density of Codes Identifying Vertices in the Square Grid
SIAM Journal on Discrete Mathematics
On a new class of codes for identifying vertices in graphs
IEEE Transactions on Information Theory
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For a graph, G, and a vertex v@?V(G), let N[v] be the set of vertices adjacent to and including v. A set D@?V(G) is a (vertex) identifying code if for any two distinct vertices v"1,v"2@?V(G), the vertex sets N[v"1]@?D and N[v"2]@?D are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of 37~0.428571, and this remains the best known upper bound. Until now, the best known lower bound was 1229~0.413793 and was proved by Cranston and Yu in 2009. We present three new codes with a density of 37, and we improve the lower bound to 512~0.416667.