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On the FAULT-TOLERANT Resolvability in Line Graphs of Dragon and Kayak Paddles Graphs

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  • Muhammad Faheem
  • Zohaib Zahid
  • Yong Chen

Abstract

Because of its wide range of applications, metric resolvability has been used in chemical structures, computer networks, and electrical circuits. It has been applied as a node (sensor) in an electric circuit. The electric circuit will not be able to flow current if one node (sensor) in that chain becomes faulty. The fault-tolerant selfstable circuit is a circuit that permits the current flow even if one of the nodes (sensors) becomes faulty. If the removal of any node from a resolving set (RS) of the circuit is still a RS, then the RS of the circuit is considered a fault-tolerant resolving set (FTRS) and the fault-tolerant metric dimension (FTMD) is its minimum cardinality. Even though the problem of finding the exact values of MD in line graphs seems to be even harder, the FTMD for the line graphs was first discussed by Guo et al. [13]. Ahmad et al. [5] determined the precise value of the MD for the line graph of the kayak paddle graph. We calculate the precise value of the FTMD for the line graph in this family of graphs. The FTMD is a more generalized invariant than the MD. We also consider the problem of obtaining a precise value for this parameter in the line graph of the dragon graph. It is concluded that these families have a constant FTMD.

Suggested Citation

  • Muhammad Faheem & Zohaib Zahid & Yong Chen, 2022. "On the FAULT-TOLERANT Resolvability in Line Graphs of Dragon and Kayak Paddles Graphs," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-9, April.
    • Handle: RePEc:hin:jnlmpe:7688702
    DOI: 10.1155/2022/7688702
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