Optimal Identifying Codes in Cycles and Paths

Junnila, Ville; Laihonen, Tero

doi:10.1007/s00373-011-1058-6

Cited by 16 publications

(11 citation statements)

References 8 publications

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“…On the other hand, the study of integer identifying codes in powers of cycles had taken several years (see e.g. [6,21,39]) before being completed by Junnila and Laihonen [28]. We have the following results.…”

Section: Cyclesmentioning

confidence: 80%

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Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

Gravier

Parreau

Rottey

et al. 2015

Electron. J. Combin.

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We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V |) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V | α with α ∈ { 1 4 , 1 3 , 2 5 }. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.

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Section: Cyclesmentioning

confidence: 80%

“…Identifying codes have already been studied in different classes of vertex-transitive graphs, especially in cycles [6,21,28,39] and hypercubes [7,12,13,27,29]. In these examples, the order of the size of an optimal identifying code seems to always match its fractional value.…”

Section: Introductionmentioning

confidence: 99%

Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

Gravier

Parreau

Rottey

et al. 2015

Electron. J. Combin.

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“…Cycles have been investigated for identifying codes [8] and r-identifying codes [7,10], as well as for locally identifying colorings [11].…”

Section: Definition 3 (Coloring)mentioning

confidence: 99%

Extension of universal cycles for globally identifying colorings of cycles

Coupechoux

2017

Discrete Mathematics

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In 1998, Karpovsky, Chakrabarty and Levitin introduced identifying codes to model fault diagnosis in multiprocessor systems [1]. In these codes, each vertex is identified by the vertices belonging to the code in its neighborhood. There exists a coloring variant as follows: a globally identifying coloring of a graph is a coloring such that each vertex is identified by the colors in its neighborhood. We aim at finding the maximum length of a cycle with such a coloring, given a fixed number of colors we can use. Parreau [2] used Jackson's work [3] on universal cycles to give a lower bound of this length. In this article, we will adapt what Jackson did, to improve this result.

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“…where the calculations are done modulo n. Previously, in [2,4,7,9,13,17,19,23], identifying and locating-dominating codes have been studied in the circulant graphs C n (1, 2, . .…”

mentioning

confidence: 99%

Optimal bounds on codes for location in circulant graphs

2018

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Identifying and locating-dominating codes have been studied widely in circulant graphs of type Cn(1, 2, 3, . . . , r) over the recent years. In 2013, Ghebleh and Niepel studied locatingdominating and identifying codes in the circulant graphs Cn(1, d) for d = 3 and proposed as an open question the case of d > 3. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs Cn(1, d), Cn(1, d − 1, d) and Cn(1, d − 1, d, d + 1). We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in Cn(1, 3) and Cn(1, 4).

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Optimal Identifying Codes in Cycles and Paths

Cited by 16 publications

References 8 publications

Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

Extension of universal cycles for globally identifying colorings of cycles

Optimal bounds on codes for location in circulant graphs

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