On finding short reconfiguration sequences between independent sets

Token Sliding Optimization asks whether there exists a sequence of at most ℓ steps that transforms independent set S into T, where at each step a token slides to an unoccupied neighboring vertex (while maintaining independence). In Token Jumping Optimization, we are instead allowed to jump from a vertex to any unoccupied vertex. Both problems are known to be FPT when parameterized by ℓ on nowhere dense graphs. In this work, we show that both problems are FPT for parameter $k+\ell +d$ on d-degenerate graphs as well as for parameter $\left|M\right|+\ell +\Delta$ on graphs having a modulator M to maximum degree Δ. We complement these results by showing that for parameter ℓ both problems become hard already on 2-degenerate graphs. Finally, we show that using such families one can obtain a unified algorithm for the standard Token Jumping problem parameterized by k on degenerate and nowhere dense graphs.