Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $(t,r)$ broadcast domination of directed graphs   A dominating set of a graph $G$ is a set of vertices that contains at least
one endpoint of every edge on the graph. The domination number of $G$ is the
order of a minimum dominating set of $G$. The $(t,r)$ broadcast domination is a
generalization of domination in which a set of broadcasting vertices emits
signals of strength $t$ that decrease by 1 as they traverse each edge, and we
require that every vertex in the graph receives a cumulative signal of at least
$r$ from its set of broadcasting neighbors. In this paper, we extend the study
of $(t,r)$ broadcast domination to directed graphs. Our main result explores
the interval of values obtained by considering the directed $(t,r)$ broadcast
domination numbers of all orientations of a graph $G$. In particular, we prove
that in the cases $r=1$ and $(t,r) = (2,2)$, for every integer value in this
interval, there exists an orientation $\vec{G}$ of $G$ which has directed
$(t,r)$ broadcast domination number equal to that value. We also investigate
directed $(t,r)$ broadcast domination on the finite grid graph, the star graph,
the infinite grid graph, and the infinite triangular lattice graph. We conclude
with some directions for future study.

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The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th