  Let $G=(V,E)$ be a digraph having no loops and no multiple arcs, with vertex
set $V=\{v_1,v_2,\ldots,v_n\}$ and arc set $E=\{e_1,e_2,\ldots,e_m\}$. Denote
the adjacency matrix and the vertex in-degree diagonal matrix of $G$ by
$A=(a_{ij})_{n\times n}$ and $D=diag(d^+(v_1),d^+(v_2),\cdots,d^+(v_n))$, where
$a_{ij}=1$ if $(v_i,v_j)\in E(G)$ and $a_{ij}=0$ otherwise, and $d^+(v_i)$ is
the number of arcs with head $v_i$. Set $f_1(G;x)=\det(xI-A),
f_2(G;x)=\det(xI-D+A),f_3(G;x)=\det(xI-D-A),f_4(G;x)={\rm per}(xI-A),
f_5(G;x)={\rm per}(xI-D+A),f_6(G;x)={\rm per}(xI-D-A)$, where $\det(X)$ and
${\rm per}(X)$ denote the determinant and the permanent of a square matrix $X$,
respectively. In this paper, we consider a variant of the Ulam's vertex
reconstruction conjecture and the Harary's edge reconstruction conjecture, and
prove that, for any $1\leq i\leq 6$, \begin{equation*}
(m-n)f_i(G;x)+xf_i'(G;x)=\sum\limits_{e\in E}f_i(G-e;x), \end{equation*}
  which implies that if $m\neq n$, then $f_i(G;x)$ can be reconstructed from
$\{f_i(G-e;x)|e\in E\}$.
