Feedback vertex set reconfiguration in planar graphs

Abstract

We study the complexity of deciding whether for two given feedback vertex sets of a graph there is a step-by-step transformation between them, such that for each feedback vertex set in the transformation, the next one is obtained by exchanging a single vertex. We give a classification of the complexity of this question for planar graphs in terms of the maximum degree. We show that for planar graphs of maximum degree at most three the problem is tractable because there always exists a transformation, while it is PSPACE-complete when the maximum degree is at most four. The positive side of the classification extends to 𝘒₃, ₃ -minor-free graphs of maximum degree three. We then consider the Matroid Parity problem, which generalizes feedback vertex sets in graphs of maximum degree three as well as matchings and spanning trees in general graphs. Generalizing known results for the latter two we show that there always exists a transformation between any two non-maximum independent parity sets.