  In this paper, we study the Reconstruction Conjecture for finite simple
graphs. Let $\Gamma$ and $\Gamma'$ be finite simple graphs with at least three
vertices such that there exists a bijective map $f:V(\Gamma) \rightarrow
V(\Gamma')$ and for any $v\in V(\Gamma)$, there exists an isomorphism
$\phi_v:\Gamma-v \to \Gamma'-f(v)$. Then we define the associated directed
graph $\widetilde{\Gamma}=\widetilde{\Gamma}(\Gamma,\Gamma',f,\{\phi_v\}_{v\in
V(\Gamma)})$ with two kinds of arrows from the graphs $\Gamma$ and $\Gamma'$,
the bijective map $f$ and the isomorphisms $\{\phi_v\}_{v\in V(\Gamma)}$. By
investigating the associated directed graph $\widetilde{\Gamma}$, we study when
are the two graphs $\Gamma$ and $\Gamma'$ isomorphic.
