Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Dulmage-Mendelsohn Decomposition for $b$-Matchings   We establish the theory of the Dulmage-Mendelsohn decomposition for
$b$-matchings. The original Dulmage-Mendelsohn decomposition is a classical
canonical decomposition of bipartite graphs, which describes the structures of
the maximum $1$-matchings and the dual optimizers, i.e., the minimum vertex
covers. In this paper, we develop analogical properties, and thus obtain the
structure of the maximum $b$-matchings and characterizes the family of
$b$-verifying set.

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The categories of this paper are: physics
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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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: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga