Linear-time algorithm for sliding tokens on trees

Suppose that we are given two independent sets IbIb and IrIr of a graph such that |Ib|=|Ir||Ib|=|Ir|, and imagine that a token is placed on each vertex in IbIb. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms IbIb into IrIr 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 IbIb and IrIr 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.