Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting matchings with k unmatched vertices in planar graphs   We consider the problem of counting matchings in planar graphs. While perfect
matchings in planar graphs can be counted by a classical polynomial-time
algorithm, the problem of counting all matchings (possibly containing unmatched
vertices, also known as defects) is known to be #P-complete on planar graphs.
To interpolate between the hard case of counting matchings and the easy case of
counting perfect matchings, we study the parameterized problem of counting
matchings with exactly k unmatched vertices in a planar graph G, on input G and
k. This setting has a natural interpretation in statistical physics, and it is
a special case of counting perfect matchings in k-apex graphs (graphs that can
be turned planar by removing at most k vertices).
  Starting from a recent #W[1]-hardness proof for counting perfect matchings on
k-apex graphs, we obtain that counting matchings with k unmatched vertices in
planar graphs is #W[1]-hard. In contrast, given a plane graph G with s
distinguished faces, there is an $O(2^s \cdot n^3)$ time algorithm for counting
those matchings with k unmatched vertices such that all unmatched vertices lie
on the distinguished faces. This implies an $f(k,s)\cdot n^{O(1)}$ time
algorithm for counting perfect matchings in k-apex graphs whose apex
neighborhood is covered by s faces.

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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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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: math-ph
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The categories of this paper are: math
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: eess