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