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Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

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In an undirected graph G = (V; E) a subset C V is called an identifying code, if the sets B1 (v) \ C consisting of all elements of C within distance one from the vertex v are nonempty and di erent. We take G to be the in nite hexagonal grid, and show that the density of any identifying code is at least 16=39 and that there is an identifying code of density 3=7.

A binary code C f0; 1g n is called r-identifying, if the sets B r ðxÞ \ C; where B r ðxÞ is the set of all vectors within the Hamming distance r from x; are all nonempty and no two are the same. Denote by M r ðnÞ the minimum possible cardinality of a binary r-identifying code in f0; 1g n : We prove that if r 2 ½0; 1Þ is a constant, then lim n!1 n À1 log 2 M b rnc ðnÞ ¼ 1 À H ðrÞ; where H ðxÞ ¼ Àx log 2 x À ð1 À xÞ log 2 ð1 À xÞ: We also prove that the problem whether or not a given binary linear code is r-identifying is P 2 -complete. # 2002 Elsevier Science (USA)

Let G=(V, E) be an undirected graph and C a subset of vertices. If the sets B r (v) 5 C, v ¥ V, are all nonempty and different, where B r (v) denotes the set of all points within distance r from v, we call C an r-identifying code. We give bounds on the best possible density of r-identifying codes in the two-dimensional square lattice. © 2002 Elsevier Science (USA)

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