Locating and identifying codes in circulant networks

Ghebleh, Mohammad; Niepel, Ludovít

doi:10.1016/j.dam.2013.03.008

Cited by 8 publications

(22 citation statements)

References 21 publications

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“…A few results have been found for closed locating-dominating sets and identifying codes in circulant graphs [32,62].…”

Section: Outline and Overviewmentioning

confidence: 99%

“…The average value of the range of u = (x, y) given that given that x < r and √r 2 − x 2 ≤ y < r, etc., is R 3 (u) = (24π−3π 2 −32)r When x < r and √ r 2 − x 2 ≤ y < r, etc., as in case 3 inFigure 9.1, the node is close to and overlaps two sides of the plane. WLOG, suppose x < r and √r 2 − x 2 ≤ y < r.The sum of the angles that overlap the x-and y− axes is θ = 2 cos −1 ( of the segments is θ 2π…”

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confidence: 99%

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Mixed-weight open locating-dominating sets

Givens

Kincaid

Mao

et al. 2017

2017 51st Annual Conference on Information Sciences and Systems (CISS)

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The detection and location of issues in a network is a common problem encompassing a wide variety of research areas. Location-detection problems have been studied for wireless sensor networks and environmental monitoring, microprocessor fault detection, public utility contamination, and finding intruders in buildings. Modeling these systems as a graph, we want to find the smallest subset of nodes that, when sensors are placed at those locations, can detect and locate any anomalies that arise. One type of set that solves this problem is the open locating-dominating set (OLD-set), a set of nodes that forms a unique and nonempty neighborhood with every node in the graph. For this work, we begin with a study of OLD-sets in circulant graphs. Circulant graphs are a group of regular cyclic graphs that are often used in massively parallel systems. We prove the optimal OLD-set size for two circulant graphs using two proof techniques: the discharging method and Hall's Theorem. Next we introduce the mixed-weight open locating-dominating set (mixed-weight OLD-set), an extension of the OLD-set. The mixed-weight OLD-set allows nodes in the graph to have different weights, representing systems that use sensors of varying strengths. This is a novel approach to the study of location-detection problems. We show that the decision problem for the minimum mixed-weight OLD-set, for any weights up to positive integer d, is NP-complete. We find the size of mixed-weight OLD-sets in paths and cycles for weights 1 and 2. We consider mixed-weight OLD-sets in random graphs by providing probabilistic bounds on the size of the mixed-weight OLD-set and use simulation to reinforce the theoretical results. Finally, we build and study an integer linear program to solve for mixed-weight OLD-sets and use greedy algorithms to generate mixed-weight OLD-set estimates in random geometric graphs. We also extend our results for mixed-weight OLD-sets in random graphs to random geometric graphs by estimating the probabilistic upper bound for the size of the set.

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“…A few results have been found for closed locating-dominating sets and identifying codes in circulant graphs [32,62].…”

Section: Outline and Overviewmentioning

confidence: 99%

mentioning

confidence: 99%

Mixed-weight open locating-dominating sets

Givens

Kincaid

Mao

et al. 2017

2017 51st Annual Conference on Information Sciences and Systems (CISS)

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“…Great efforts are made in studying locating and identifying codes in circulants. M. Ghebleh and L. Niepel studied locating and identifying code numbers of C n (1,3) in [11], but they stated as an open question what happens in the graphs C n (1, d) with d being greater than 3 and mentioned that the methods used in their paper seem to be nonapplicable. So, Ville Junnila, Tero Laihonen, and Gabrielle Paris present a new approach to study identifying, locating-dominating, and self-identifying codes in the circulant graphs…”

Section: Introductionmentioning

confidence: 99%

Locating and Identifying Codes in Circulant Graphs

Song

Zhang

2021

Discrete Dynamics in Nature and Society

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Identifying and locating-dominating codes have been studied widely in circulant graphs. Recently, Ville Junnila et al. (Optimal bounds on codes for location in circulant graphs, Cryptography and Communications; 2019) studied identifying and locating-dominating codes in circulants C n 1 , d , C n 1 , d − 1 , d , and C n 1 , d − 1 , d , d + 1 . In this paper, identifying, locating, and self-identifying codes in the circulant graphs C n k , d , C n k , d − k , d , and C n k , d − k , d , d + k are studied, and this extends Junnila et al.’s results to general cases.

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“…A locating-dominating set (LDS) in a connected graph G = (V, E) is a dominating set S of G such that for every pair of vertices u and v in V (G) \ S, N (u) ∩ S = N (v) ∩ S. The minimum cardinality of a locating-dominating set of G is called the locating-domination number γ L (G) [1]. The locating-domination problem has been discussed for paths and cycles [4,5], infinite grids [6], circulant graphs [7], fault-tolerant graphs [8] and so on.…”

Section: Introductionmentioning

confidence: 99%