Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar.
We study problems of reconfigurability of independent sets in graphs. We consider three different models (token jumping, token sliding, and token addition and removal) and analyze relationships between them. We prove that independent set reconfigurability in perfect graphs (under any of the three models) generalizes the shortest path reconfigurability problem in general graphs and is therefore PSPACE-complete. On the positive side, we give polynomial results for even-hole-free graphs and P"4-free graphs. Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. We demonstrate that a host of reconfiguration problems derived from NP-complete problems are PSPACE-complete, while some are also NP-hard to approximate. In contrast, several reconfiguration versions of problems in P are solvable in polynomial time.