## Abstract

Let G=(V(G),E(G))

be a simple connected unweighted graph. A set R⊂V(G)

is called a fault-tolerant resolving set with the tolerance level k if the cardinality of the set Sx,y={w∈R:d(w,x)≠d(w,y)}

is at least k for every pair of distinct vertices x,y

of G. A k-level metric dimension refers to the minimum size of a fault-tolerant resolving set with the tolerance level k. In this article, we calculate and determine the k-level metric dimension for the circulant graph C(n:1,2)

for all possible values of k and n.

The optimal fault-tolerant resolving sets with k tolerance are also delineated.

be a simple connected unweighted graph. A set R⊂V(G)

is called a fault-tolerant resolving set with the tolerance level k if the cardinality of the set Sx,y={w∈R:d(w,x)≠d(w,y)}

is at least k for every pair of distinct vertices x,y

of G. A k-level metric dimension refers to the minimum size of a fault-tolerant resolving set with the tolerance level k. In this article, we calculate and determine the k-level metric dimension for the circulant graph C(n:1,2)

for all possible values of k and n.

The optimal fault-tolerant resolving sets with k tolerance are also delineated.

Original language | English |
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Article number | 1896 |

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Mathematics |

Volume | 11 |

Issue number | 8 |

DOIs | |

Publication status | Published - 17 Apr 2023 |