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A graph $G$ with a perfect matching is called saturated if $G+e$ has moreperfect matchings than $G$ for any edge $e$ that is not in $G$. Lov\'asz gave acharacterization of the saturated graphs called the cathedral theorem, withsome applications to the enumeration problem of perfect matchings, and laterSzigeti gave another proof. In this paper, we give a new proof with ourpreceding works which revealed canonical structures of general graphs withperfect matchings. Here, the cathedral theorem is derived in quite a naturalway, providing more refined or generalized properties. Moreover, the new proofshows that it can be proved without using the Gallai-Edmonds structure theorem.

The Third Proof of Lov\'asz's Cathedral Theorem