Linear-Time Algorithm for Sliding Tokens on Trees

Suppose that we are given two independent sets $Ib$ and $Ir$ of a graph such that $|Ib|=|Ir|$, and imagine that a token is placed on each vertex in $Ib$. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms $Ib$ into $Ir$ 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 paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between $Ib$ and $I_r$ whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.