Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Computing upper bounds for optimal density of $(t,r)$ broadcasts on the
  infinite grid   The domination number of a finite graph $G$ with vertex set $V$ is the
cardinality of the smallest set $S\subseteq V$ such that for every vertex $v\in
V$ either $v\in S$ or $v$ is adjacent to a vertex in $S$. A set $S$ satisfying
these conditions is called a dominating set. In 2015 Blessing, Insko, Johnson,
and Mauretour introduced $(t,r)$ broadcast domination, a generalization of
graph domination parameterized by the nonnegative integers $t$ and $r$. In this
setting, we say that the signal a vertex $v\in V$ receives from a tower of
strength $t$ located at vertex $T$ is defined by $sig(v,T)=max(t-dist(v,T),0)$.
Then a $(t,r)$ broadcast dominating set on $G$ is a set $S\subseteq V$ such
that the sum of all signal received at each vertex $v \in V$ is at least $r$.
In this paper, we consider $(t,r)$ broadcasts of the infinite grid and present
a Python program to compute upper bounds on the minimal density of a $(t,r)$
broadcast on the infinite grid. These upper bounds allow us to construct
counterexamples to a conjecture by Blessing et al. that the $(t,r)$ and $(t+1,
r+2)$ broadcasts are equal whenever $t,r\geq 1$.

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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: physics