2021/06/27 by S. Prabhu, V. Manimozhi, Prabhu, S. +5 · 1 citation
Computer Science · Engineering · #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graphene and Nanomaterials Applications
paper · pdf · doi:10.48550/arxiv.2106.14285
openalex publication_date 2021/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of vertices S⊆ V(G) is a basis or 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 basis S is a fault-tolerant basis if S∖ \x\ is a basis for every x ∈ S. The fault-tolerant metric dimension (FTMD) β'(G) of G is the minimum cardinality of a fault-tolerant basis. It is shown that each twin vertex of G belongs to every fault-tolerant basis 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.