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.

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.