Edge version of metric dimension for the families of grid graphs and generalized prism graphs

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Monitoring and controlling complex networks is of great importance to understand different types of technological and physical systems for source localization. Source localization refers to the process of determining the location or position of a signal source in space based on measurements obtained from multiple sensors. Doubly resolving sets, also known as doubly-resolving arrays, are a particular type of sensor configuration that can enhance the accuracy of source localization. In other words, source localization in a network is equivalent to calculating minimal doubly resolving sets (mDRS) in a network. The concept of the minimal edge version of doubly resolving sets (evDRS) is extension of mDRS. In this article, we take into account the optimization problem of locating the evDRSs for the classes of generalized prism and grid networks. Also, it is demonstrated that the evDRSs for the classes of generalized prisms and grid networks have constant cardinality. This research presents a novel approach with implications for complex network structures such as, network security and communication systems. Furthermore, the findings may have broader implications for diverse fields such as sensor networks, telecommunications, and distributed computing, where prism and grid-like structures are prevalent. The suggested approach may help to improve network optimization and facilitate more robust and reliable grid-based systems.INDEX TERMS Grid networks, Generalized prism networks, Line networks, Doubly resolving sets, Edge computing.

“…In [25], the class of G α,γ was studied for evMD. The result of evMD for the class of G α,γ networks is given as:…”

Section: Edge Version Of Doubly Resolving Sets Of the Class Of Grid N...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Edge Version of Doubly Resolving Sets for Grid and Generalized Prism Networks

Nasir,

Ahmad,

Zahid

et al. 2024

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“…Metric dimensions of nanotubes VC 5 C 7 and H-naphthalene are discussed in [34]. The n-sunlit and prism graphs are explained in [35], they have constant edge metric dimensions. Computation of metric and edge metric dimension of different types of windmill graph in [36].…”

Section: Introductionmentioning

confidence: 99%

Honeycomb Rhombic Torus Vertex-Edge Based Resolvability Parameters and Its Application in Robot Navigation

Bukhari,

Jamil,

Azeem

et al. 2024

In the aircraft sector, honeycomb composite materials are frequently employed. Recent research has demonstrated the benefits of honeycomb structures in applications involving nanohole arrays in anodized alumina, micro-porous arrays in polymer thin films, activated carbon honeycombs, and photonic band gap honeycomb structures. The resolvability parameter is the area of graph theory that is most commonly explored. This results in an original network reconfiguration. Occasionally in terms of atoms (metric dimension), and sometimes in terms of bounds (edge metric dimension). In this article, we examined the honeycomb rhombic torus's metric, edge metric, mixed metric, and partition dimension. The application of edge metric dimension is also discussed in it.INDEX TERMS Edge metric dimension, honeycomb rhombic torus, metric dimension, mixed metric dimension, partition dimension, resolving set.

“…J.B. Liu et al considered the family of necklace graphs for the edge version of metric dimension in [29]. Also, in [33] R. Nasir et al discussed the edge version of metric dimension for the families of grid graphs and generalized prism graphs. In metric dimension, the resolving sets are referred as detecting devices in computer network problems while in the edge version of metric dimension we consider the edge version of resolving sets for the same purpose.…”

Section: Introductionmentioning

confidence: 99%

The Edge Version of Metric Dimension for the Family of Circulant Graphs Cₙ(1, 2)

Nasir

et al. 2021

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Graph theory is widely used to analyze the structure models in chemistry, biology, computer science, operations research and sociology. Molecular bonds, species movement between regions, development of computer algorithms, shortest spanning trees in weighted graphs, aircraft scheduling and exploration of diffusion mechanisms are some of these structure models. Let G = (V G , E G ) be a connected graph, where V G and E G represent the set of vertices and the set of edges respectively. The idea of the edge version of metric dimension is based on the distance of edges in a graph. Let R E G be the smallest set of edges in a connected graph G that forms a basis such that for every pair of edges e 1 , e 2 ∈ E G , there exists an edge e ∈ R E G for which d E G (e 1 , e) = d E G (e 2 , e) holds. In this paper, we show that the family of circulant graphs C n (1, 2) is the family of graphs with constant edge version of metric dimension.INDEX TERMS Line graph, Resolving sets, The edge version of metric dimension, Circulant graphs.