Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual
  Carving-Width   For a clustered graph, i.e, a graph whose vertex set is recursively
partitioned into clusters, the C-Planarity Testing problem asks whether it is
possible to find a planar embedding of the graph and a representation of each
cluster as a region homeomorphic to a closed disk such that 1. the subgraph
induced by each cluster is drawn in the interior of the corresponding disk, 2.
each edge intersects any disk at most once, and 3. the nesting between clusters
is reflected by the representation, i.e., child clusters are properly contained
in their parent cluster. The computational complexity of this problem, whose
study has been central to the theory of graph visualization since its
introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades.
Planarity for clustered graphs. ESA'95], has only been recently settled
[Radoslav Fulek and Csaba D. T\'oth. Atomic Embeddability, Clustered Planarity,
and Thickenability. To appear at SODA'20]. Before such a breakthrough, the
complexity question was still unsolved even when the graph has a prescribed
planar embedding, i.e, for embedded clustered graphs.
  We show that the C-Planarity Testing problem admits a single-exponential
single-parameter FPT algorithm for embedded clustered graphs, when
parameterized by the carving-width of the dual graph of the input. This is the
first FPT algorithm for this long-standing open problem with respect to a
single notable graph-width parameter. Moreover, in the general case, the
polynomial dependency of our FPT algorithm is smaller than the one of the
algorithm by Fulek and T\'oth. To further strengthen the relevance of this
result, we show that the C-Planarity Testing problem retains its computational
complexity when parameterized by several other graph-width parameters, which
may potentially lead to faster algorithms.

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The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
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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: mtrl-th
12
The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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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: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: math