Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Third Proof of Lov\'asz's Cathedral Theorem   A graph $G$ with a perfect matching is called saturated if $G+e$ has more
perfect matchings than $G$ for any edge $e$ that is not in $G$. Lov\'asz gave a
characterization of the saturated graphs called the cathedral theorem, with
some applications to the enumeration problem of perfect matchings, and later
Szigeti gave another proof. In this paper, we give a new proof with our
preceding works which revealed canonical structures of general graphs with
perfect matchings. Here, the cathedral theorem is derived in quite a natural
way, providing more refined or generalized properties. Moreover, the new proof
shows that it can be proved without using the Gallai-Edmonds structure theorem.

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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin