Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sufficient conditions for a digraph to admit a
  $(1,\leq\ell)$-identifying code   A $(1,\le \ell)$-identifying code in a digraph $D$ is a subset $C$ of
vertices of $D$ such that all distinct subsets of vertices of cardinality at
most $\ell$ have different closed in-neighborhoods within $C$. In this paper,
we give some sufficient conditions for a digraph of minimum in-degree
$\delta^-\ge 1$ to admit a $(1,\le \ell)$-identifying code for $\ell=\delta^-,
\delta^-+1$. As a corollary, we obtain the result by Laihonen that states that
a graph of minimum degree $\delta\ge 2$ and girth at least 7 admits a $(1,\le
\delta)$-identifying code. Moreover, we prove that every $1$-in-regular digraph
has a $(1,\le 2)$-identifying code if and only if the girth of the digraph is
at least 5. We also characterize all the 2-in-regular digraphs admitting a
$(1,\le \ell)$-identifying code for $\ell=2,3$.

19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph