🤖 AI Summary
This work addresses the problem of determining the minimum number of sliding moves required to reconfigure one pair of non-adjacent labeled vertices on a graph into another such pair. Focusing on the two-token independent set reconfiguration problem, the authors combine graph-theoretic analysis with combinatorial reasoning to devise path-construction and state-transition strategies. They establish, for the first time, that in any connected component of the reconfiguration graph induced by $n$-vertex graphs, any feasible reconfiguration between two valid states can be accomplished in at most $4n$ steps. This result provides a linear upper bound, resolving an open question posed by Briański et al. at MFCS 2021 and marking a significant theoretical advance in the study of token reconfiguration problems.
📝 Abstract
We consider the problem of shifting two tokens placed on nonadjacent vertices $u,v$ of a graph $G$ on $n$ vertices to two nonadjacent vertices $u',v'$ of $G$ using a sequence of token movements. In each step, a token is moved from the vertex it is on to a neighbour of that vertex, ensuring that the tokens remain on nonadjacent vertices after this move. We answer a question of Briański, Felsner, Hodor, and Micek [``Reconfiguring Independent Sets on Interval Graphs'', MFCS 2021] by showing that if the two tokens can be moved from their initial position to their final position, then it can be done using at most $4n$ moves.