bibkey: fresanfigueroa2026mutualvisibility
authors: Julián A. Fresán-Figueroa, Ilán A. Goldfeder, Aldo Lozano-Piña
year: 2026
title: "Reconfiguration graphs of mutual-visibility sets"
doi: 10.48550/arXiv.2610.24324
url: https://arxiv.org/html/2610.24324v1
claim: "The question after Theorem 3.6 asks whether the full mixed reconfiguration graph of mutual-visibility sets is maximally connected for every connected host."
strata_touched:
  - D5/S3/Combinatorics/Graph/MutualVisibility/Model
  - D5/S3/Combinatorics/Graph/MutualVisibility/Geodesics
license: citation-only
triage: anchor

Mixed reconfiguration of mutual-visibility sets

The primary source is arXiv:2610.24324v1. Section 2 defines a mutual-visibility set by the existence, for each pair of distinct selected vertices, of a shortest path in the original host whose internal vertices avoid the selected set. The path may depend on the pair. Empty sets and all feasible ranks are retained. Graphs are finite, simple, undirected and connected unless stated otherwise.

Proposition 2.1 states that every subset of a mutual-visibility set is again a mutual-visibility set. It attributes this hereditary property to S. Cicerone, G. Di Stefano, L. Drožđek, J. Hedžet, S. Klavžar and I. G. Yero, “Variety of mutual-visibility problems in graphs”, Theoretical Computer Science 974 (2023), 114096, DOI 10.1016/j.tcs.2023.114096. The attribution and bibliographic details are given in the primary source’s Proposition 2.1 and reference list.

Two distinct feasible sets are adjacent when their symmetric difference has size one, or their two set differences are singletons whose vertices are adjacent in the host. Thus token addition/removal and actual host-edge slides are both present. Fixed-cardinality layers and slide-only token graphs are different objects.

The question immediately after Theorem 3.6, repeated in Section 7, asks whether the full reconfiguration graph is maximally connected for every connected host: its vertex connectivity must equal its minimum degree.

The source’s preceding order-at-least-three bound is not a hypothesis of this question. Host orders one and two remain in scope. The finite separator convention is that deleting a cut leaves at most one vertex or disconnects the remaining induced graph. It gives connectivity zero for a singleton complete graph and order minus one for every nonempty complete graph.

The formal model uses finite subsets and original-host shortest walks; minimum-length walks are paths. Strict mandatory ancestry, used in the rooted geometry, requires membership on every root geodesic. This auxiliary relation is stronger than membership on some shortest path and does not replace existential mutual visibility. The full connectivity equality is proved by D5/S3/Combinatorics/Graph/MutualVisibility/MinimumPair.result, using rooted comparability and simultaneous cut payments in addition to the model and inward-replacement geometry. Its problem dossier states the full conclusion and its fixed-layer boundary.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2610.24324
• Primary version: https://arxiv.org/html/2610.24324v1