  As a variant of the Ulam's vertex reconstruction conjecture and the Harary's
edge reconstruction conjecture, Cvetkovi\'c and Schwenk posed independently the
following problem: Can the characteristic polynomial of a simple graph $G$ with
vertex set $V$ be reconstructed from the characteristic polynomials of all
subgraphs in $\{G-v|v\in V\}$ for $|V|\geq 3$? This problem is still open. A
natural problem is: Can the characteristic polynomial of a simple graph $G$
with edge set $E$ be reconstructed from the characteristic polynomials of all
subgraphs in $\{G-e|e\in E\}$? In this paper, we prove that if $|V|\neq |E|$,
then the characteristic polynomial of $G$ can be reconstructed from the
characteristic polynomials of all subgraphs in $\{G-uv, G-u-v|uv\in E\}$, and
the similar result holds for the permanental polynomial of $G$. We also prove
that the Laplacian (resp. signless Laplacian) characteristic polynomial of $G$
can be reconstructed from the Laplacian (resp. signless Laplacian)
characteristic polynomials of all subgraphs in $\{G-e|e\in E\}$ (resp. if
$|V|\neq |E|$).
