Locating–dominating codes in paths

Exoo, Geoffrey; Junnila, Ville; Laihonen, Tero

doi:10.1016/j.disc.2011.05.004

Cited by 10 publications

(28 citation statements)

References 12 publications

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“…Finding these bounds will help determine best practices for fault detection and design in networked microprocessors. locating-dominating sets have been studied for paths and cycles [8,15,25,26,39,44,70], hypercubes [45,46,47,51,68], and grids [50,66,76]. The mixed-weight OLD-set is also similar to an OLD-set on a directed graph: increased weight can be represented by arcs to other nodes.…”

Section: Discussionmentioning

confidence: 99%

“…In and cycles [8,15,25,26,39,44,70]. We will study the mixed-weight OLD-set in paths and cycles with a focus on weights 1 and 2.…”

Section: Discussionmentioning

confidence: 99%

“…The mixed-weight OLD-set provides the foundation for the theoretical study of networked sensors of different strengths. Related problems have been studied for systems with stronger than normal sensors [8,15,25,26,39,45,47,50,51,66,68,70,76], but this is the first time strong sensors have been studied in OLD-sets and the first time varied strength sensors have been considered for any location-detection problems. Our contributions include a theoretical introduction to mixed-weight OLD-sets and their properties, results in a variety of graphs, establishing an integer linear program to solve for the sets, and examining greedy algorithms to estimate the sets in large graphs.…”

Section: Contributionsmentioning

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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Section: Discussionmentioning

confidence: 99%

“…In and cycles [8,15,25,26,39,44,70]. We will study the mixed-weight OLD-set in paths and cycles with a focus on weights 1 and 2.…”

Section: Discussionmentioning

confidence: 99%

Section: Contributionsmentioning

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 problem of finding an optimal locating-dominating code in arbitrary network is NPcomplete [2]. Optimal locating codes for special classes of graphs were studied in [1,3,[5][6][7]18]. Locating-dominating sets in infinite grids and their density were studied in [14,15,17,19].…”

Section: Introductionmentioning

confidence: 99%

Locating–paired-dominating sets in square grids

Niepel

2015

Discrete Mathematics

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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

Self Cite

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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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Locating–dominating codes in paths

Cited by 10 publications

References 12 publications

Mixed-weight open locating-dominating sets

Mixed-weight open locating-dominating sets

Locating–paired-dominating sets in square grids

Optimal bounds on codes for location in circulant graphs

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