Abstract
The concept of resolving sets (RSs) and metric dimension (MD) invariants have a wide range of applications in robot navigation, computer networks, and chemical structure. RS has been used as a sensor in an indoor positioning system to find an interrupter. Many terminologies in machine learning have also been used to diagnose the interrupter in the systems of marine and gas turbines using sensory data. We proposed a fault‐tolerant self‐stable system that allows for the detection of an interrupter even if one of the sensors in the chain fails. If the elimination of any element from a RS is still a RS, then the RS is considered as a fault‐tolerant resolving set (FTRS), and the fault‐tolerant metric dimension (FTMD) is its minimum cardinality. In this paper, we calculated the FTMD of the subdivision graphs of the necklace and prism graphs. We also found that this invariant has constant values for both graphs.
Highlights
Introduction and PreliminariesLet G be a connected, undirected, finite, and simple graph with vertex set V(G) and edge set E(G). e number of edges in the shortest x − y path between two vertices x, y ∈ V(G) is known as the distance dG(x, y) between them. e cardinality of edges that are incident to a vertex x is called degree dG(x) of x
E concept of resolving sets (RSs) and metric dimension (MD) invariants have a wide range of applications in robot navigation, computer networks, and chemical structure
We proposed a fault-tolerant self-stable system that allows for the detection of an interrupter even if one of the sensors in the chain fails
Summary
E concept of resolving sets (RSs) and metric dimension (MD) invariants have a wide range of applications in robot navigation, computer networks, and chemical structure. RS has been used as a sensor in an indoor positioning system to find an interrupter. Many terminologies in machine learning have been used to diagnose the interrupter in the systems of marine and gas turbines using sensory data. We proposed a fault-tolerant self-stable system that allows for the detection of an interrupter even if one of the sensors in the chain fails. If the elimination of any element from a RS is still a RS, the RS is considered as a fault-tolerant resolving set (FTRS), and the fault-tolerant metric dimension (FTMD) is its minimum cardinality. We calculated the FTMD of the subdivision graphs of the necklace and prism graphs. We found that this invariant has constant values for both graphs
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