A leaf matching operation on a graph consists of removing vertex degree 1 together with its neighbour from the graph. Let G be d-regular cyclically ( d − + 2 k ) -edge-connected even order, where ≥ 0 and 3. We prove that for any given set X edges, there is no 1-factor avoiding if only either an isolated can obtained by series operations in X, or has independent contains more than half vertices ...

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 pol...