Reconfiguration graphs of mutual-visibility sets

Source abstract

A set $S\subseteq V(G)$ is a mutual-visibility set if every pair of vertices in $S$ is joined by a geodesic whose internal vertices do not belong to $S$. We define a reconfiguration graph whose vertices are the mutual-visibility sets of $G$. Two configurations are adjacent if one is obtained from the other by adding or removing one vertex or by sliding one token along an edge of $G$. The subgraphs induced by configurations of a fixed cardinality are called layers. The spanning subgraph formed by token-addition and token-removal edges is a daisy cube. We study the effects of token-sliding edges on distances and connectivity to this skeleton. For a connected host of order $n\ge3$, the vertex connectivity of the reconfiguration graph lies between $3$ and $n$, and both bounds are attained. We determine several parameters of $\mathcal{R}_{\mathcal{M}}(K_n)$ and describe the behavior of the model for disconnected hosts and for isometric or convex subgraphs. We also give a component-based criterion for mobile mutual visibility. For complete bipartite graphs, we use a description of feasible configurations to determine connectivity and distance bounds for fixed-cardinality layers. For trees, we show that every component of a layer of cardinality at least three is isomorphic to a Cartesian product of subtrees, obtaining exact metric formulas and characterizing the connected layers. Finally, we prove that every layer of a cycle is connected.