Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sparse Hop Spanners for Unit Disk Graphs   A unit disk graph $G$ on a given set $P$ of points in the plane is a
geometric graph where an edge exists between two points $p,q \in P$ if and only
if $|pq| \leq 1$. A spanning subgraph $G'$ of $G$ is a $k$-hop spanner if and
only if for every edge $pq\in G$, there is a path between $p,q$ in $G'$ with at
most $k$ edges. We obtain the following results for unit disk graphs in the
plane.
  (I) Every $n$-vertex unit disk graph has a $5$-hop spanner with at most
$5.5n$ edges. We analyze the family of spanners constructed by Biniaz (2020)
and improve the upper bound on the number of edges from $9n$ to $5.5n$.
  (II) Using a new construction, we show that every $n$-vertex unit disk graph
has a $3$-hop spanner with at most $11n$ edges.
  (III) Every $n$-vertex unit disk graph has a $2$-hop spanner with $O(n\log
n)$ edges. This is the first nontrivial construction of $2$-hop spanners.
  (IV) For every sufficiently large positive integer $n$, there exists a set
$P$ of $n$ points on a circle, such that every plane hop spanner on $P$ has hop
stretch factor at least $4$. Previously, no lower bound greater than $2$ was
known.
  (V) For every finite point set on a circle, there exists a plane (i.e.,
crossing-free) $4$-hop spanner. As such, this provides a tight bound for points
on a circle.
  (VI) The maximum degree of $k$-hop spanners cannot be bounded from above by a
function of $k$ for any positive integer $k$.

28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph