Minimum Separator Reconfiguration

We study the problem of reconfiguring one minimum s-t-separator A into another minimum s-tseparator B in some n-vertex graph G containing two non-adjacent vertices s and t. We consider several variants of the problem as we focus on both the token sliding and token jumping models. Our first contribution is a polynomial-time algorithm that computes (if one exists) a minimum-length sequence of slides transforming A into B. We additionally establish that the existence of a sequence of jumps (which need not be of minimum length) can be decided in polynomial time (by an algorithm that also outputs a witnessing sequence when one exists). In contrast, and somewhat surprisingly, we show that deciding if a sequence of at most l jumps can transform A into B is an NP-complete problem. To complement this negative result, we investigate the parameterized complexity of what we believe to be the two most natural parameterized counterparts of the latter problem; in particular, we study the problem of computing a minimum-length sequence of jumps when parameterized by the size k of the minimum s-t-separators and when parameterized by the number l of jumps. For the first parameterization, we show that the problem is fixed-parameter tractable, but does not admit a polynomial kernel unless NP ? coNP/poly. We complete the picture by designing a kernel with O(l2) vertices and edges for the length l of the sequence as a parameter.