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I am a physicist who is interested in the applications of graph theory.

I've been studying the bipartite graphs and perfect matching finding problems. I see there are several research works on algorithms for finding perfect matchings when a bigraph is given. However, I could not find any research about "construcing bigraphs from a set of perfect matchings".

My question is as follows: For a given set of perfect matchings $A$ with the same number of vertices, we can easily construct a bigraph which gives a set of perfect matchings $B$ that includes $A$ ($B \supseteq A$). Then, what is the condition for $A$ so that $B=A$?

Is there any mathematical paper related to the problem? Thank you in advance.

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  • $\begingroup$ In other words, when any perfect matching in $\cup A$ is listed in $A$? Which type of answer do you expect? $\endgroup$
    – Fedor Petrov
    Commented Mar 26, 2023 at 14:37
  • $\begingroup$ Yeah, I think you just rephrased my question. I just want to know the conditions of A that can fulfill such a special property, or whether there exists any research on the subject. $\endgroup$
    – Beom.Jean
    Commented Mar 26, 2023 at 17:00

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