Total mutual visibility in Fibonacci and Lucas cubes
A set of vertices is total mutual-visible if every pair of graph vertices has a shortest path whose internal vertices avoid the set. We prove that the total mutual-visibility numbers of the n-dimensional Fibonacci and Lucas cubes both have order ρⁿ/n, where ρ > 1 satisfies ρ³ = ρ² + 1. The eligible Fibonacci words are exactly those avoiding 11 for which zero padding avoids 0000. Their induced graph decomposes into fibres of recognizing subsets of maximal independent sets of a path, and each fibre is a Cartesian product of smaller Fibonacci cubes. Total visibility becomes open packing in…
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