The Third Proof of Lovász's Cathedral Theorem

This paper is on matching theory. A graph with perfect matching is called saturated if any addition of one complement edge creates a new perfect matching. Lovász gave a characterization of the saturated graphs called the cathedral theorem, and later Szigeti gave another proof. In this paper, we gave a new proof with our preceding works, which revealed the canonical structure of general graph with perfect matchings. Here, the cathedral theorem is derived in quite a natural way and shows that it holds independently from the Gallai-Edmonds structure theorem.