The reconstruction conjecture for finite simple graphs and associated directed graphs

Abstract

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)→V(Γ′) and for any v∈V(Γ), there exists an isomorphism ϕv:Γ−v→Γ′−f(v). Then we define the associated directed graph Γ˜=Γ˜(Γ,Γ′,f,{ϕv}v∈V(Γ)) with two kinds of arrows from the graphs Γ and Γ′, the bijective map f and the isomorphisms {ϕv}v∈V(Γ). By investigating the associated directed graph Γ˜, we study when are the two graphs Γ and Γ′ isomorphic.