Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extending Precolorings to Distinguish Group Actions   Given a group $\Gamma$ acting on a set $X$, a $k$-coloring
$\phi:X\to\{1,\dots,k\}$ of $X$ is distinguishing with respect to $\Gamma$ if
the only $\gamma\in \Gamma$ that fixes $\phi$ is the identity action. The
distinguishing number of the action $\Gamma$, denoted $D_{\Gamma}(X)$, is then
the smallest positive integer $k$ such that there is a distinguishing
$k$-coloring of $X$ with respect to $\Gamma$. This notion has been studied in a
number of settings, but by far the largest body of work has been concerned with
finding the distinguishing number of the action of the automorphism group of a
graph $G$ upon its vertex set, which is referred to as the distinguishing
number of $G$.
  The distinguishing number of a group action is a measure of how difficult it
is to "break" all of the permutations arising from that action. In this paper,
we aim to further differentiate the resilience of group actions with the same
distinguishing number. In particular, we introduce a precoloring extension
framework to address this issue. A set $S \subseteq X$ is a fixing set for
$\Gamma$ if for every non-identity element $\gamma \in \Gamma$ there is an
element $s \in S$ such that $\gamma(s) \neq s$. The distinguishing extension
number $\operatorname{ext}_D(X,\Gamma;k)$ is the minimum number $m$ such that
for all fixing sets $W \subseteq X$ with $|W| \geq m$, every $k$-coloring $c :
X \setminus W \to [k]$ can be extended to a $k$-coloring that distinguishes
$X$.
  In this paper, we prove that
$\operatorname{ext}_D(\mathbb{R},\operatorname{Aut}(\mathbb{R}),2) =4$, where
$\operatorname{Aut}(\mathbb{R})$ is comprised of compositions of translations
and reflections. We also consider the distinguishing extension number of the
circle and (finite) cycles, obtaining several exact results and bounds.

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The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th