Token Sliding on Chordal Graphs

Let I be an independent set of a graph G. Imagine that a token is located on any vertex of I. We can now move the tokens of I along the edges of the graph as long as the set of tokens still defines an independent set of G. Given two independent sets I and J, the Token Sliding problem consists in deciding whether there exists a sequence of independent sets which transforms I into J so that every pair of consecutive independent sets of the sequence can be obtained via a token move. This problem is known to be PSPACE-complete even on planar graphs. In 2014, Demaine et al. asked whether the Token Sliding reconfiguration problem is polynomial time solvable on interval graphs and more generally in chordal graphs. Yamada and Uehara showed in 2016 that a polynomial time transformation can be found in proper interval graphs. In this paper, we answer the first question of Demaine et al. and generalize the result of Yamada and Uehara by showing that we can decide in polynomial time whether two independent sets of an interval graph are in the same connected component. Moveover, we answer similar questions by showing that: (i) determining if there exists a token sliding transformation between every pair of k-independent sets in an interval graph can be decided in polynomial time; (ii) deciding this problem becomes co-NP-hard and even co-W[2]-hard (parameterized by the size of the independent set) on split graphs, a sub-class of chordal graphs.