Metric Dimension of Generalized Möbius Ladder and its Application to WSN Localization
Muhammad Idrees*, Hongbin Ma*,, Mei Wu*, Abdul Rauf Nizami**, Mobeen Munir**, and Sajid Ali***
*School of Automation, Beijing Institute of Technology
No.5 South Zhongguancun Street, Haidian District, Beijing 100081, China
**Abdus Salam School of Mathematical Sciences, Government College University Lahore
68-B, New Muslim Town, Lahore, Punjab 54600, Pakistan
***Department of Computer Science, University of Education
DG Khan Campus, Kangan Road, DG Khan, Punjab, Pakistan
Localization is one of the key techniques in wireless sensor network. While the global positioning system (GPS) is one of the most popular positioning technologies, the weakness of high cost and energy consuming makes it difficult to install in every node. In order to reduce the cost and energy consumption only a few nodes, called beacon nodes, are equipped with GPS modules. The remaining nodes obtain their locations through localization. In order to find the minimum positions of beacons, a resolving set with minimal cardinality has been obtained in the network which is called metric basis. Simultaneous local metric basis of the network is also given in which each pair of adjacent vertices of the network is distinguished by some element of simultaneous local metric basis which makes the network design more reasonable. In this paper a new network, the generalized Möbius ladder Mm,n, has been introduced and its metric dimension and simultaneous local metric dimension of its two subfamilies have been calculated.
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