Polynomial-Time Algorithm for Sliding Tokens on Trees

Suppose that we are given two independent sets

I

$$_{b}$$

and

I

$$_{r}$$

of a graph such that

$$\mid $$

$${{\varvec{I}}}_{b}$$

$$\mid $$

=

$$\mid $$

I

$$_{r}$$

$$\mid $$

, 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

$$_{b}$$

and

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 paper, we show that the problem is solvable for trees in quadratic time. Our proof is constructive: for a yes-instance, we can find an actual sequence of independent sets between

I

$$_{b}$$

and

I

$$_{r}$$

whose length (i.e., the number of token-slides) is quadratic. We note that there exists an infinite family of instances on paths for which any sequence requires quadratic length.