Fault-Tolerant Metric Dimension of Cube of Paths
Laxman Saha
Abstract
Abstract For a simple connected graph G = ( V ( G ) , E ( G )), a set R ⊆ V ( G ) is said to be a resolving set of G if every pair of vertices of G are resolved by some vertices in R i.e., every pair of vertices of G are identified uniquely by some vertex elements in F . A resolving set of G containing the minimum number of vertices is the metric basis and the minimum cardinality of the metric basis is called the metric dimension of G . A resolving set F for the graph G is said to be fault tolerant if for each u ∈ F , F \ { u } is also a resolving set for G and the minimum cardinality of the fault-tolerant resolving set said to be the fault-tolerant metric dimension . In this article, we determine the exact value of fault-tolerant metric dimension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> .