Instances requiring repeated visits by both tokens

Determine whether there exist two-token sliding reconfiguration instances in which every valid sequence transforming the initial independent set into the target independent set requires both tokens to visit a large number of vertices more than once.

Background

The paper studies how often vertices must be visited during a token-sliding reconfiguration of two-token independent sets. It gives examples where one token must revisit many vertices, while also proving that a reconfiguration can always be found in which no vertex is visited more than four times in total.

The authors explicitly note that they do not know an example forcing both tokens individually to revisit a large number of vertices. Thus, the unresolved issue concerns whether the repeated-visit phenomenon can necessarily affect both tokens simultaneously in some construction.