Optimal Fault-Tolerant Resolving Set of Power Paths

Laxman Saha, Rupen Lama, Bapan Das, Avishek Adhikari and Kinkar Chandra Das ()
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Laxman Saha: Department of Mathematics, Balurghat College, Balurghat 733101, India
Rupen Lama: Department of Mathematics, Balurghat College, Balurghat 733101, India
Bapan Das: Department of Mathematics, Balurghat College, Balurghat 733101, India
Avishek Adhikari: Department of Mathematics, Presidency University, Kolkata 700073, India
Kinkar Chandra Das: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

Mathematics, 2023, vol. 11, issue 13, 1-18

Abstract: In a simple connected undirected graph G , an ordered set R of vertices is called a resolving set if for every pair of distinct vertices u and v , there is a vertex w ∈ R such that d ( u , w ) ≠ d ( v , w ) . A resolving set F for the graph G is a fault-tolerant resolving set if for each v ∈ F , F ∖ { v } is also a resolving set for G . In this article, we determine an optimal fault-resolving set of r -th power of any path P n when n ≥ r ( r − 1 ) + 2 . For the other values of n , we give bounds for the size of an optimal fault-resolving set. We have also presented an algorithm to construct a fault-tolerant resolving set of P m r from a fault-tolerant resolving set of P n r where m < n .

Keywords: resolving set; metric dimension; fault-tolerant metric dimension; power of path (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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