The Reconstruction Conjecture for finite simple graphs and associated directed graphs

In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ be finite simple graphs with at least three vertices such that there exists a bijective map $f:V(\Gamma) \rightarrow V(\Gamma’)$andforany$v\in V(\Gamma)$,thereexistsanisomorphism$\phi_v:\Gamma-v \to \Gamma’-f(v)$. Then we define the associated directed graph $\widetilde{\Gamma}=\widetilde{\Gamma}(\Gamma,\Gamma’,f,{\phi_v}_{v\in V(\Gamma)})$withtwokindsofarrowsfromthegraphs$\Gamma$and$\Gamma’$, the bijective map $f$ and the isomorphisms $\{{\varphi }_v{\}}_{v\in V(\mathrm{\Gamma })}$. By investigating the associated directed graph $\tilde{\mathrm{\Gamma }}$, we study when are the two graphs $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ isomorphic.