Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complexity Results in Graph Reconstruction   We investigate the relative complexity of the graph isomorphism problem (GI)
and problems related to the reconstruction of a graph from its vertex-deleted
or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate
deck (LD) problems). We show that these problems are closely related for all
amounts $c \geq 1$ of deletion:
  1) $GI \equiv^{l}_{iso} VDC_{c}$, $GI \equiv^{l}_{iso} EDC_{c}$, $GI
\leq^{l}_{m} LVD_c$, and $GI \equiv^{p}_{iso} LED_c$.
  2) For all $k \geq 2$, $GI \equiv^{p}_{iso} k-VDC_c$ and $GI \equiv^{p}_{iso}
k-EDC_c$.
  3) For all $k \geq 2$, $GI \leq^{l}_{m} k-LVD_c$.
  4)$GI \equiv^{p}_{iso} 2-LVC_c$.
  5) For all $k \geq 2$, $GI \equiv^{p}_{iso} k-LED_c$.
  For many of these results, even the $c = 1$ case was not previously known.
  Similar to the definition of reconstruction numbers $vrn_{\exists}(G)$ [HP85]
and $ern_{\exists}(G)$ (see page 120 of [LS03]), we introduce two new graph
parameters, $vrn_{\forall}(G)$ and $ern_{\forall}(G)$, and give an example of a
family $\{G_n\}_{n \geq 4}$ of graphs on $n$ vertices for which
$vrn_{\exists}(G_n) < vrn_{\forall}(G_n)$. For every $k \geq 2$ and $n \geq 1$,
we show that there exists a collection of $k$ graphs on $(2^{k-1}+1)n+k$
vertices with $2^{n}$ 1-vertex-preimages, i.e., one has families of graph
collections whose number of 1-vertex-preimages is huge relative to the size of
the graphs involved.

17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
2
The categories of this paper are: math
24
The categories of this paper are: dg-ga
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
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The categories of this paper are: stat
5
The categories of this paper are: gr-qc