Polynomial-Time Algorithms for Sliding Tokens on Cactus Graphs and Block Graphs

Given two independent sets I, J of a graph G, and imagine that a token (coin) is placed at each vertex of I. The Sliding Token problem asks if one could transform I to J via a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors so that the resulting set of vertices where tokens are placed remains independent. This problem is PSPACE-complete even for planar graphs of maximum degree 3 and boundedtreewidth. In this paper, we show that Sliding Token can be solved efficiently for cactus graphs and block graphs, and give upper bounds on the length of a transformation sequence between any two independent sets of these graph classes. Our algorithms are designed based on two main observations. First, all structures that forbid the existence of a sequence of token slidings between I and J , if exist, can be found in polynomial time. A sufficient condition for determining no-instances can be easily derived using this characterization. Second, without such forbidden structures, a sequence of token slidings between I and J does exist. In this case, one can indeed transform I to J (and vice versa) using a polynomial number of token-slides.