Suppose that we are given two independent sets $I_b$ and $I_r$ of a graph such that $|I_b|=|I_r|$, and imagine that a token is placed on each vertex in $I_b$. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms $I_b$ into $I_r$ so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this talk, we thus study the problem restricted to trees, and give the following three results:
(Based on a paper from ISAAC 2014 and Theoretical Computer Science 2015 by Demaine, Demaine, Fox-Epstein, Hoang, Ito, Ono, Otachi, Uehara, and Yamada.)