A Study on Minimal Doubly Resolving Sets of Certain Families of Networks

Ahmad, Muhammad; Zahid, Zohaib; Javaid, Muhammad; Ashebo, Mamo Abebe

doi:10.1109/access.2023.3282701

Cited by 3 publications

(1 citation statement)

References 35 publications

(36 reference statements)

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“…The minimal DRS of a few convex polytopes, such as prisms and hamming networks, has been investigated in [27], [28], and [29], respectively. Recently in another research, Ahmad et al [30] arrived at generalized solutions for the DRS problem of some families of networks. Further, the DRSs of Cartesian products G × H is given a lower and an upper bound by Nie and Xu [31], who also generalize the previously established finding regarding P 2 × C n .…”

Section: Introductionmentioning

confidence: 99%

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

Nasir,

Ahmad,

Zahid

et al. 2024

IEEE Access

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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.

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Section: Introductionmentioning

confidence: 99%

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

Nasir,

Ahmad,

Zahid

et al. 2024

IEEE Access

Self Cite

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Optimizing SNARK networks via double metric dimension

Ahmad,

Faheem,

Bajri

et al. 2024

MATH

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<p>Doubly resolving sets (DRSs) provide a promising approach for source detection. They consist of minimal subsets of nodes with the smallest cardinality, referred to as the double metric dimension (DMD), that can uniquely identify the location of any other node within the network. Utilizing DRSs can improve the accuracy and efficiency of the identification of the origin of a diffusion process. This ability is crucial for early intervention and control in scenarios such as epidemic outbreaks, misinformation spreading in social media, and fault detection in communication networks. In this study, we computed the DMD of flower snarks $ J_{m} $ and quasi-flower snarks $ G_{m} $ by describing their minimal doubly resolving sets (MDRSs). We deduce that the DMD for the flower snarks $ J_{m} $ is finite and depends on the network's order, and the DMD for the quasi-flower snarks $ G_{m} $ is finite and independent of the network's order. Furthermore, our findings offer valuable insights into the structural features of complex networks. This knowledge can offer direction for future studies in network theory and its practical implementations.</p>

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Characterizing edge-based doubly resolving sets within circulant networks $ C_n(1, 2) $

Nasir,

Ahmad,

Zahid

et al. 2024

MATH

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<abstract><p>The focus of this article lies on the notion of the edge version of doubly resolving sets (EVDRSs) in circulant networks. EVDRSs refer to unique edge subsets that are necessary for identifying individual edges in a network and distinguishing them based on their edge distances to the elements of the EVDRS. The main objectives were to define the minimal size of EVDRSs for circulant networks $ C_n(1, 2) $ and to investigate their basic properties. The systematic research helped to achieve a new understanding of the existence, construction, and characterization of EVDRSs in circulant networks $ C_n(1, 2) $. It is established that the EVDRSs in the circulant network $ C_n(1, 2) $ are finite and are bounded by the order of the network. Among the numerous implications of these findings are those that refer to the design and optimization of distributed sensor networks, improving communication and network protocols, as well as tracking the spread of infectious diseases and epidemics over social networks. The application of the identified methodology helps improve the process of network optimization which contributes to the development of more effective and robust circulant-based structures.</p></abstract>

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A Study on Minimal Doubly Resolving Sets of Certain Families of Networks

Cited by 3 publications

References 35 publications

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

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

Optimizing SNARK networks via double metric dimension

Characterizing edge-based doubly resolving sets within circulant networks $ C_n(1, 2) $

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