On Fault-Tolerant Resolvability of Double Antiprism and its related Graphs

2021/04/19 by Sharma, Sunny Kumar, Bhat, Vijay Kumar
#05C12 #68R01 #68R10 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.09167

Abstract

For a connected graph Γ=(V,E), a subset R of ordered vertices in V is said to be a resolving set in Γ, if the vector of distances to the vertices in R is unique for each uⁱ∈ V(Γ). The metric dimension of Γ is the minimum cardinality of such a set R. If R∖ \uⁱ\ is still a resolving set ∀ uⁱ∈ R, then R is called a fault-tolerant resolving set (FTRS) for Γ and its least cardinality is the fault-tolerant metric dimension (FTMD) of Γ. In this article, we introduce the concept of an independent fault-tolerant resolving set (IFTRS) and investigate it for several well-known graphs. We also show that the FTMD is four for three closely related families of convex polytopes available in the literature (viz., double antiprism \mathbbA_n, S_n, and T_n).

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