On Codes Identifying Vertices in the Two-Dimensional Square Lattice with Diagonals
IEEE Transactions on Computers
Fault-tolerant locating-dominating sets
Discrete Mathematics
Identifying and locating-dominating codes on chains and cycles
European Journal of Combinatorics
On a new class of identifying codes in graphs
Information Processing Letters
On a new class of codes for identifying vertices in graphs
IEEE Transactions on Information Theory
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A subset C@?V is an r-identifying code in a graph G=(V,E) if the sets I"r(v)={c@?C|d(c,v)@?r} are distinct and non-empty for all vertices v@?V. Here, d(c,v) denotes the number of edges on any shortest path from c to v. We consider the infinite n-dimensional king grid, i.e., the graph with vertex set V=Z^n and the edge set E={{x=(x"1,...,x"n),y=(y"1,...,y"n)}||x"i-y"i|@?1for i=1,...,n,xy}, and give some lower bounds on the density of an r-identifying code. In particular, we prove that for n=3 and for all r=15, the optimal density of an r-identifying code is 18r^2. The problem finding a minimum identifying code in the 3-dimensional king grid is equivalent with a minimum packing problem of cubes in the 3-dimensional lattice so that every point is covered by a distinct and non-empty subset of cubes.