The reconstruction conjecture for finite simple graphs and associated directed graphs

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