Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parameterized Lower Bounds and Dichotomy Results for the NP-completeness
  of $H$-free Edge Modification Problems   For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist
at most $k$ edges whose deletion from the input graph $G$ results in a graph
without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge
Editing are defined similarly where only completion (addition) of edges are
allowed in the former and both completion and deletion are allowed in the
latter. We completely settle the classical complexities of these problems by
proving that $H$-free Edge Deletion is NP-complete if and only if $H$ is a
graph with at least two edges, $H$-free Edge Completion is NP-complete if and
only if $H$ is a graph with at least two non-edges and $H$-free Edge Editing is
NP-complete if and only if $H$ is a graph with at least three vertices.
Additionally, we prove that, these NP-complete problems cannot be solved in
parameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$,
unless Exponential Time Hypothesis fails. Furthermore, we obtain implications
on the incompressibility of these problems.

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The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: math
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph