Shortest Reconfiguration of Perfect Matchings via Alternating Cycles

Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of matchings via alternating cycles. Namely, want to find a sequence which transforms one given another such that symmetric difference each pair consecutive is single cycle. The equivalent combinatorial path polytopes. We prove NP-hard even when graph planar or bipartite, but it can be solved polynomial time outerplanar.