Token Jumping in minor-closed classes

TL;DR

This paper extends fixed-parameter tractability results for the Token Jumping reconfiguration problem from $K_{3, ext{ extellipsis}}$-free graphs to all $K_{ ext{ extellipsis}}, ext{ extellipsis}$-free graphs, including minor-free classes, showing broader applicability.

Contribution

The authors generalize the FPT algorithm for Token Jumping reconfiguration from $K_{3, ext{ extellipsis}}$-free graphs to all $K_{ ext{ extellipsis}}, ext{ extellipsis}$-free graphs, expanding the classes where the problem is efficiently solvable.

Findings

Token Jumping is FPT on $K_{ ext{ extellipsis}}, ext{ extellipsis}$-free graphs.

The result applies to many well-known graph classes, including minor-free graphs.

Deciding reconfiguration in these classes is computationally feasible.

Abstract

Given two $k$-independent sets $I$ and $J$ of a graph $G$, one can ask if it is possible to transform the one into the other in such a way that, at any step, we replace one vertex of the current independent set by another while keeping the property of being independent. Deciding this problem, known as the Token Jumping (TJ) reconfiguration problem, is PSPACE-complete even on planar graphs. Ito et al. proved in 2014 that the problem is FPT parameterized by $k$ if the input graph is $K_{3,\ell}$-free. We prove that the result of Ito et al. can be extended to any $K_{\ell,\ell}$-free graphs. In other words, if $G$ is a $K_{\ell,\ell}$-free graph, then it is possible to decide in FPT-time if $I$ can be transformed into $J$. As a by product, the TJ-reconfiguration problem is FPT in many well-known classes of graphs such as any minor-free class.