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I might be missing something, but it looks like you want a partition of the edges of $K_n$, odd $n$, into a set of subgraphs which are regular of degree 2. That's called a 2-factorization. It can be …

I'll just address the "up to isomorphism" requirement, assuming you have an algorithm for making all the perfect matchings. I'm assuming you are only consider embedding-preserving automorphisms. A con …

Take a circular ladder of odd length. Also known as the cartesian product of $C_n$ ($n$ odd) and $K_2$. For every perfect matching there are many ways to change it into a different perfect matching by …

Counterexample for general planar bipartite graphs I'll take "the planar bipartite graph is generated by 2-cells which are 4-cycles" to mean that every edge lies on at least one 4-face. Consider a c …

If the edge weights are scaled by a sufficiently high factor, a minimum weight matching will have the least greatest weight. By also removing the edges with weight less than $w$, a minimum weight matc …

All simple non-empty regular graphs of even degree have a two factor, see here . So you are just asking when they have a 1-factor. In addition to having an even number of vertices, the conditions are …

Yes, it is still just as hard. Given an arbitrary bipartite graph, in polynomial time you can remove every edge that is not in a perfect matching (test one edge at a time), thus reducing the problem …

One infinite family is the lexicographic product (composition) $C_s[\bar K_t]$, where $s$ is odd and $t$ is even. Basically you take the cycle $C_s$ and expand each vertex into a cluster of $t$ indepe …