Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Characterizing extremal graphs for open neighbourhood
  location-domination   An open neighbourhood locating-dominating set is a set $S$ of vertices of a
graph $G$ such that each vertex of $G$ has a neighbour in $S$, and for any two
vertices $u,v$ of $G$, there is at least one vertex in $S$ that is a neighbour
of exactly one of $u$ and $v$. We characterize those graphs whose only open
neighbourhood locating-dominating set is the whole set of vertices. More
precisely, we prove that these graphs are exactly the graphs all whose
connected components are half-graphs (a half-graph is a special bipartite graph
with both parts of the same size, where each part can be ordered so that the
open neighbourhoods of consecutive vertices differ by exactly one vertex). This
corrects a wrong characterization from the literature.

3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga