Galactic Token Sliding

TL;DR

This paper introduces a new galactic reconfiguration model for independent set reconfiguration, demonstrating fixed-parameter tractability of token sliding on various graph classes, thus advancing understanding of this problem's complexity.

Contribution

The paper proposes the galactic reconfiguration model and shows fixed-parameter tractability of token sliding on bounded degree, planar, and chordal graphs, filling gaps in existing research.

Findings

Token sliding is fixed-parameter tractable on bounded degree graphs.

Token sliding is fixed-parameter tractable on planar graphs.

Token sliding is fixed-parameter tractable on chordal graphs with bounded clique number.

Abstract

Given a graph $G$ and two independent sets $I_s$ and $I_t$ of size $k$, the independent set reconfiguration problem asks whether there exists a sequence of $k$-sized independent sets $I_s = I_0, I_1, I_2, \ldots, I_\ell = I_t$ such that each independent set is obtained from the previous one using a so-called reconfiguration step. Viewing each independent set as a collection of $k$ tokens placed on the vertices of a graph $G$, the two most studied reconfiguration steps are token jumping and token sliding. In the token jumping variant of the problem, a single step allows a token to jump from one vertex to any other vertex in the graph. In the token sliding variant, a token is only allowed to slide from a vertex to one of its neighbors. Like the independent set problem, both of the aforementioned problems are known to be W[1]-hard on general graphs. A very fruitful line of research has showed that the independent set problem becomes fixed-parameter tractable when restricted to sparse graph classes, such as planar, bounded treewidth, nowhere-dense, and all the way to biclique-free graphs. Over a series of papers, the same was shown to hold for the token jumping problem. As for the token sliding problem, which is mentioned in most of these papers, almost nothing is known beyond the fact that the problem is polynomial-time solvable on trees and interval graphs. We remedy this situation by introducing a new model for the reconfiguration of independent sets, which we call galactic reconfiguration. Using this new model, we show that (standard) token sliding is fixed-parameter tractable on graphs of bounded degree, planar graphs, and chordal graphs of bounded clique number. We believe that the galactic reconfiguration model is of independent interest and could potentially help in resolving the remaining open questions concerning the (parameterized) complexity of token sliding.