Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks

S. Prabhu, V. Manimozhi, M. Arulperumjothi and Sandi Klavžar

Applied Mathematics and Computation, 2022, vol. 420, issue C

Abstract: A set of vertices S⊆V(G) is a resolving set of a graph G if for each x,y∈V(G) there is a vertex u∈S such that d(x,u)≠d(y,u). A resolving set S is a fault-tolerant resolving set if S∖{x} is a resolving set for every x∈S. The fault-tolerant metric dimension (FTMD) β′(G) of G is the minimum cardinality of a fault-tolerant resolving set. It is shown that each twin vertex of G belongs to every fault-tolerant resolving set of G. As a consequence, β′(G)=n(G) iff each vertex of G is a twin vertex, which corrects a wrong characterization of graphs G with β′(G)=n(G) from [Mathematics 7(1) (2019) 78]. This FTMD problem is reinvestigated for Butterfly networks, Benes networks, and silicate networks. This extends partial results from [IEEE Access 8 (2020) 145435–145445], and at the same time, disproves related conjectures from the same paper.

Keywords: Metric dimension; Fault-tolerant metric dimension; Twin vertex; Benes network; Butterfly network; Silicate network (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:420:y:2022:i:c:s0096300321009802

DOI: 10.1016/j.amc.2021.126897

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