Are Bipartite Matching and General Matching Really Different Problems? 
Questions:

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*Have there been attempts to either prove or disprove, that every general matching problem can be transformed into a bipartite matching problem, from whose solution the solution of the original problem can determined efficiently?


*How would a proof of the existence or non-existence of such a transformation affect matching theory?

The reason for asking is that apparently the time complexity of finding minimum weight perfect bipartite matchings equals that of minimum weight perfect general matching, which seems strange in view of the different techniques that are used for solving the problems and, while implementations for weighted bipartite matching abound, implementations for general weighted matching are very rare.
 A: In the unweighted setting, Nobert Blum's paper "A new approach to maximum matching in general graphs.(1990)" introduced the idea of transforming general graph G to a bipartite graph G' and computing the maximum matching by applying Hopcroft-Karp style algorithm on G'. The paper had some bugs which prompted the author to rewrite the proof in (1999) and again in 2015 "Maximum matching in general graphs without explicit consideration of blossoms revisited. (2015)". To the best of my knowledge, this is the only paper (in unweighted setting) which tries to solve general matching by converting it to bipartite matching problem.
The complexity of general graph matching algorithms stems from the odd length alternating cycles which are absent in bipartite graphs. A better framework to simplify the handling of these odd length cycles would help in bridging the gap between the 2 classes of graphs in matching theory. 
I am not sure of any developments in this regard for weighted general matching which is a  tougher problem than maximum cardinality general matching. Hope this gives some heads up to future visitors.
