Minimum separator reconfiguration

We study the problem of reconfiguring s-t-separators on finite simple graphs. We consider several variants of the problem, focusing on the token sliding and jumping models. We begin with a polynomial-time algorithm that computes (if one exists) a shortest sequence of slides and another that decides if a sequence of jumps exists and outputs a witnessing sequence. We also show that deciding if a reconfiguration sequence of at most ℓ jumps exists is an NP-complete problem. To complement this result, we investigate the parameterized complexity of the natural parameterizations of the problem: by the size k of the minimum s-t-separators and by the number of jumps ℓ. We show that the problem is in FPT parameterized by k, but that it does not admit a polynomial kernel unless NP⊆coNP/poly. Our final result is a kernel with O(ℓ²) vertices and edges.