An Optimal Edge-Robust Identifying Code in the Triangular Lattice

Honkala, Iiro

doi:10.1007/s00026-004-0222-6

Cited by 13 publications

(6 citation statements)

References 13 publications

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“…That means that on average, the number of vertices of C in a frame u + F is at least 16 5 . Using the same method as in the proof of Proposition 1 (see [4] and [9]), we can conclude that the density of C is at least 16 5 × 1 |F | = 4 15 . We can improve this lower bound by further analysis and more advanced discharging rules.…”

Section: Casementioning

confidence: 83%

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Tolerant identification with Euclidean balls

2012

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The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this paper, we consider as sensors the set Z 2 where one sensor can check its neighbours within Euclidean distance r. We construct tolerant identifying codes in this network that are robust against some changes in the neighbourhood monitored by each sensor. We give bounds for the smallest density of a tolerant identifying code for general values of r. We also provide infinite families of values r with optimal such codes and study the case of small values of r.

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Section: Casementioning

confidence: 83%

“…As we will see in the next section, for infinite family of (r, ∆) there are optimal codes of density 1 2 so we cannot expect in this case to have a general upper bound of order better than a constant. 4 Better constructions for given values of (r, ∆)…”

Section: Upper Boundmentioning

confidence: 99%

Tolerant identification with Euclidean balls

2012

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“…Identifying codes and locating-dominating codes have been extensively studied: see the Internet bibliography [11] maintained by Antoine Lobstein. For results on the triangular grid, see, e.g., [1], [2], [4], [6] and [5].…”

Section: Introductionmentioning

confidence: 99%

An Optimal Strongly Identifying Code in the Infinite Triangular Grid

Honkala¹

2010

Electron. J. Combin.

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Assume that $G = (V, E)$ is an undirected graph, and $C \subseteq V$. For every ${\bf v} \in V$, we denote by $I({\bf v})$ the set of all elements of $C$ that are within distance one from ${\bf v}$. If the sets $I({\bf v})\setminus \{{\bf v}\}$ for ${\bf v}\in V$ are all nonempty, and, moreover, the sets $\{ I({\bf v}), I({\bf v}) \setminus \{ {\bf v}\}\}$ for ${\bf v} \in V$ are disjoint, then $C$ is called a strongly identifying code. The smallest possible density of a strongly identifying code in the infinite triangular grid is shown to be $6/19$.

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“…In this paper we consider the following two classes of robust identifying codes from [16]. For other types of robust identifying codes, see also [14], [11], [12], [13], [17], [19], [20], [21], [22], [26].…”

Section: Introductionmentioning

confidence: 99%

On Identifying Codes in the King Grid that are Robust Against Edge Deletions

Honkala¹,

Laihonen²

2008

Electron. J. Combin.

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Assume that $G = (V, E)$ is an undirected graph, and $C \subseteq V$. For every $v \in V$, we denote $I_r(G;v) = \{ u \in C: d(u,v) \leq r\}$, where $d(u,v)$ denotes the number of edges on any shortest path from $u$ to $v$. If all the sets $I_r(G;v)$ for $v \in V$ are pairwise different, and none of them is the empty set, the code $C$ is called $r$-identifying. If $C$ is $r$-identifying in all graphs $G'$ that can be obtained from $G$ by deleting at most $t$ edges, we say that $C$ is robust against $t$ known edge deletions. Codes that are robust against $t$ unknown edge deletions form a related class. We study these two classes of codes in the king grid with the vertex set ${\Bbb Z}^2$ where two different vertices are adjacent if their Euclidean distance is at most $\sqrt{2}$.

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An Optimal Edge-Robust Identifying Code in the Triangular Lattice

Cited by 13 publications

References 13 publications

Tolerant identification with Euclidean balls

Tolerant identification with Euclidean balls

An Optimal Strongly Identifying Code in the Infinite Triangular Grid

On Identifying Codes in the King Grid that are Robust Against Edge Deletions

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