Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Degree lists and connectedness are $3$-reconstructible for graphs with
  at least seven vertices   The $k$-deck of a graph is the multiset of its subgraphs induced by $k$
vertices. A graph or graph property is $l$-reconstructible if it is determined
by the deck of subgraphs obtained by deleting $l$ vertices. We show that the
degree list of an $n$-vertex graph is $3$-reconstructible when $n\ge7$, and the
threshold on $n$ is sharp. Using this result, we show that when $n\ge7$ the
$(n-3)$-deck also determines whether an $n$-vertex graph is connected; this is
also sharp. These results extend the results of Chernyak and Manvel,
respectively, that the degree list and connectedness are $2$-reconstructible
when $n\ge6$, which are also sharp.

16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
1
The categories of this paper are: cs
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci