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Identifying Codes with Small Radius in Some Infinite Regular Graphs
"... Let G = (V; E) be a connected undirected graph and S a subset of vertices. If for all vertices v 2 V , the sets B r (v) \ S are all nonempty and different, where B r (v) denotes the set of all points within distance r from v, then we call S an ridentifying code. We give constructive upper bounds ..."
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Cited by 31 (4 self)
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Let G = (V; E) be a connected undirected graph and S a subset of vertices. If for all vertices v 2 V , the sets B r (v) \ S are all nonempty and different, where B r (v) denotes the set of all points within distance r from v, then we call S an ridentifying code. We give constructive upper bounds on the best possible density of ridentifying codes in four in finite regular graphs, for small values of r.
Exact minimum density of codes identifying vertices in the square grid
 SIAM J. Discrete Math
"... Abstract. An identifying code C is a subset of the vertices of the square grid Z 2 with the property that for each element v of Z 2, the collection of elements from C at a distance of at most one from v is nonempty and distinct from the collection of any other vertex. We prove that the minimum densi ..."
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Cited by 22 (0 self)
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Abstract. An identifying code C is a subset of the vertices of the square grid Z 2 with the property that for each element v of Z 2, the collection of elements from C at a distance of at most one from v is nonempty and distinct from the collection of any other vertex. We prove that the minimum density of C within Z 2 is 7
New Bounds for Codes Identifying Vertices in Graphs
 Electronic Journal of Combinatorics
, 1999
"... Let G = (V; E) be an undirected graph. Let C be a subset of vertices that we shall call a code. For any vertex v 2 V , the neighbouring set N(v; C) is the set of vertices of C at distance at most one from v. We say that the code C identifies the vertices of G if the neighbouring sets N(v; C); v 2 V ..."
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Cited by 17 (7 self)
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Let G = (V; E) be an undirected graph. Let C be a subset of vertices that we shall call a code. For any vertex v 2 V , the neighbouring set N(v; C) is the set of vertices of C at distance at most one from v. We say that the code C identifies the vertices of G if the neighbouring sets N(v; C); v 2 V; are all nonempty and different. What is the smallest size of an identifying code C ? We focus on the case when G is the twodimensional square lattice and improve previous upper and lower bounds on the minimum size of such a code. AMS subject classification: 05C70, 68R10, 94B99, 94C12. Submitted: February 12, 1999; Accepted: March 15, 1999. G. Cohen, A. Lobstein and G. Z'emor are with ENST and CNRS URA 820, Computer Science and Network Dept., Paris, France, I. Honkala is with Turku University, Mathematics Dept., Turku, Finland the electronic journal of combinatorics 6 (1999), #R19 2 1 Introduction In this paper, we investigate a problem initiated in [3]: given an undirected graph G = (...
On Dynamic Identifying Codes
, 2004
"... A walk c1 , c2, ..., cM in an undirected graph G =(V,E) is called a dynamic identifying code, if all the sets I(v)={u C : d(u, v) 1} for v V are nonempty and no two of them are the same set. Here d(u, v) denotes the number of edges on any shortest path from u to v,and {c1 , c2,...,cM }. We ..."
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Cited by 17 (3 self)
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A walk c1 , c2, ..., cM in an undirected graph G =(V,E) is called a dynamic identifying code, if all the sets I(v)={u C : d(u, v) 1} for v V are nonempty and no two of them are the same set. Here d(u, v) denotes the number of edges on any shortest path from u to v,and {c1 , c2,...,cM }. We consider
Structural properties of twinfree graphs
 Electronic Journal of Combinatorics
, 2007
"... Consider a connected undirected graph G = (V, E), a subset of vertices C ⊆ V, and an integer r ≥ 1; for any vertex v ∈ V, let Br(v) denote the ball of radius r centered at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V, the sets Br(v) ∩ C are al ..."
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Cited by 12 (1 self)
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Consider a connected undirected graph G = (V, E), a subset of vertices C ⊆ V, and an integer r ≥ 1; for any vertex v ∈ V, let Br(v) denote the ball of radius r centered at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V, the sets Br(v) ∩ C are all nonempty and different, then we call C an ridentifying code. A graph admits at least one ridentifying code if and only if it is rtwinfree, that is, the sets Br(v), v ∈ V, are all different. We study some structural problems in rtwinfree graphs, such as the existence of the path with 2r + 1 vertices as a subgraph, or the consequences of deleting one vertex. Given a connected, undirected, finite graph G = (V, E) and an integer r ≥ 1, we define Br(v), the ball of radius r centered at v ∈ V, by Br(v) = {x ∈ V: d(x, v) ≤ r}, where d(x, v) denotes the number of edges in any shortest path between v and x.
On Strongly Identifying Codes
 Discrete Math
, 2001
"... Identifying codes are designed for locating faulty processors in multiprocessor systems. In this paper we consider a natural extension of this problem and introduce strongly identifying codes. Several lower bounds and constructions are given and relations between dierent types of identifying codes a ..."
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Cited by 9 (2 self)
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Identifying codes are designed for locating faulty processors in multiprocessor systems. In this paper we consider a natural extension of this problem and introduce strongly identifying codes. Several lower bounds and constructions are given and relations between dierent types of identifying codes are examined.
Open neighborhood locatingdominating sets
 AUSTRALASIAN JOURNAL OF COMBINATORICS VOLUME 46 (2010), PAGES 109–119
, 2010
"... For a graph G that models a facility, various detection devices can be placed at the vertices so as to identify the location of an intruder such as a thief or saboteur. Here we introduce the open neighborhood locatingdominating set problem. This deals with problems in which the intruder at a vertex ..."
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Cited by 6 (1 self)
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For a graph G that models a facility, various detection devices can be placed at the vertices so as to identify the location of an intruder such as a thief or saboteur. Here we introduce the open neighborhood locatingdominating set problem. This deals with problems in which the intruder at a vertex can interfere with the detection device located there. We seek a minimum cardinality vertex set S with the property that for each vertex v its open neighborhood N(v) has a unique nonempty intersection with S. Such a set is an OLDset for G. Among other things, we describe minimum density OLDsets for various infinite grid graphs.
Disjoint identifyingcodes for arbitrary graphs
 In Proc. International Symposium on Information Theory (ISIT
, 2005
"... Identifying codes have been used in a variety of applications, including sensorbased location detection in harsh environments. The sensors used in such applications are typically battery powered making energy conservation an important optimization criterion for lengthening network lifetime. In this ..."
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Cited by 4 (0 self)
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Identifying codes have been used in a variety of applications, including sensorbased location detection in harsh environments. The sensors used in such applications are typically battery powered making energy conservation an important optimization criterion for lengthening network lifetime. In this work we propose and develop the concept of disjoint identifying codes with the motivation of providing energy loadbalancing in such systems. We also provide informationtheoretic upper and lower bounds on the number of disjoint identifying codes in a given graph, and show that these bounds are asymptotically tight for a modification of Hadamard matrices. A version of this paper should be presented at the IEEE Symposium on Information on Information Theory 2005. I.
Families of Optimal Codes for Strong Identification
, 2001
"... Codes for strong identification are considered. The motivation for these codes comes from locating faulty processors in a multiprocessor system. Constructions and lower bounds on these codes are given. In particular, we provide two infinite families of optimal strongly identifying codes, which can l ..."
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Codes for strong identification are considered. The motivation for these codes comes from locating faulty processors in a multiprocessor system. Constructions and lower bounds on these codes are given. In particular, we provide two infinite families of optimal strongly identifying codes, which can locate up to two malfunctioning processors in a binary hypercube.
Identifying Codes in qary Hypercubes
"... Let q be any integer ≥ 2. In this paper, we consider the qary ndimensional cube whose vertex set is Z n q and two vertices (x1,..., xn) and (y1,..., yn) are adjacent if their Lee distance is 1. As a natural extension of identifying codes in binary Hamming spaces, we further study identifying codes ..."
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Let q be any integer ≥ 2. In this paper, we consider the qary ndimensional cube whose vertex set is Z n q and two vertices (x1,..., xn) and (y1,..., yn) are adjacent if their Lee distance is 1. As a natural extension of identifying codes in binary Hamming spaces, we further study identifying codes in the above qary hypercube. We let M q t (n) denote the smallest cardinality of tidentifying codes of length n in Z n q. Little is known about ternary or quaternary identifying codes. It is known [2, 14] that M 2 1(n) ≥