Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $k$-strong induced arboricity of a graph   The induced arboricity of a graph $G$ is the smallest number of induced
forests covering the edges of $G$. This is a well-defined parameter bounded
from above by the number of edges of $G$ when each forest in a cover consists
of exactly one edge. Not all edges of a graph necessarily belong to induced
forests with larger components. For $k\geq 1$, we call an edge $k$-valid if it
is contained in an induced tree on $k$ edges. The $k$-strong induced arboricity
of $G$, denoted by $f_k(G)$, is the smallest number of induced forests with
components of sizes at least $k$ that cover all $k$-valid edges in $G$. This
parameter is highly non-monotone.
  However, we prove that for any proper minor-closed graph class $\mathcal{C}$,
and more generally for any class of bounded expansion, and any $k \geq 1$, the
maximum value of $f_k(G)$ for $G \in \mathcal{C}$ is bounded from above by a
constant depending only on $\mathcal{C}$ and $k$. This implies that the
adjacent closed vertex-distinguishing number of graphs from a class of bounded
expansion is bounded by a constant depending only on the class. We further
prove that $f_2(G) \leq 3\binom{t+1}{3}$ for any graph $G$ of tree-width~$t$
and that $f_k(G) \leq (2k)^d$ for any graph of tree-depth $d$. In addition, we
prove that $f_2(G) \leq 310$ when $G$ is planar.

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The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-ex