Galactic Token Sliding

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.

💡 Research Summary

The paper addresses the parameterized complexity of the Token Sliding (TS) variant of the Independent Set Reconfiguration problem, a setting where a token placed on a vertex may move only to an adjacent vertex while maintaining an independent set of fixed size k. While the related Token Jumping (TJ) problem has been shown to be fixed‑parameter tractable (FPT) on many sparse graph classes, TS has resisted similar treatment: apart from trees and interval graphs, no positive results were known.

Main contribution – Galactic Reconfiguration.
The authors introduce a novel abstraction called galactic graphs. The vertex set is partitioned into planets (ordinary vertices) and black holes (special vertices that may host arbitrarily many tokens). A black hole behaves like a “sink” that absorbs any number of tokens, and its neighbourhood is defined as the union of the neighbourhoods of the vertices it replaces. The key reduction rule shows that if there exists a long shortest path that lies at distance two from both the initial and target independent sets, the whole path can be replaced by a single black hole without affecting reconfigurability. Repeated application of this rule yields a kernel whose number of planets is bounded by a function f(k). Consequently, on graphs of bounded degree, TS becomes FPT (Theorem 1.1).

Types and multi‑component reduction.
To handle graph classes where long shortest paths may not exist (e.g., planar graphs), the authors develop a second tool: the notion of type. For a vertex set X, each connected component of G − X receives a type that records how the component is attached to X (the sequence of neighbours, multiplicities, etc.). If more than k·|X| + 1 components share the same type, one of them can be safely removed while preserving the existence of a reconfiguration sequence. This type‑based component reduction is a generic kernelisation technique that can be combined with the galactic reduction.

Applications.

1. Planar graphs (Theorem 1.2).
   In a large planar graph either a long shortest path exists (handled by the galactic rule) or the graph contains many vertex‑disjoint (x, y)‑paths or a vertex with many neighbours on an induced path. Both situations trigger the type‑based reduction, eventually reducing the instance to bounded degree, after which Theorem 1.1 applies. This yields an FPT algorithm for TS on planar graphs, a result that was previously open even for outerplanar graphs.

2. Chordal graphs with bounded clique number (Theorem 1.3).
   Using the clique‑tree representation of chordal graphs, the authors argue that either a long path appears in the tree (again invoking the galactic rule) or there is a vertex that appears in a large fraction of the bags along a long tree path. In the latter case, the vertex can be used to delete many bags, shrinking the graph. This leads to an FPT algorithm for TS on chordal graphs whose maximal cliques have size bounded by a function of k.

3. Hardness on split graphs (Theorem 1.4).
   A reduction from Multicolored Independent Set shows that TS remains W