1 research outputs found
Polynomial-Time Algorithms for Sliding Tokens on Cactus Graphs and Block Graphs
Given two independent sets of a graph , and imagine that a token
(coin) is placed at each vertex of . The Sliding Token problem asks if one
could transform to 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 -complete even for planar graphs of maximum degree
and bounded-treewidth. 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 and , 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 and does exist. In this case, one can indeed
transform to (and vice versa) using a polynomial number of
token-slides.Comment: The algorithm for block graphs in this manuscript contains some
flaws. More precisely, Proposition 20 is not correct. Therefore, we withdraw
this manuscrip