Edge reconstruction conjecture for finite graphs

Determine whether every finite graph with at least four edges is uniquely determined up to isomorphism by its edge deck ED(G), defined as the multiset of all subgraphs obtained by deleting a single edge from G.

Background

The paper studies reconstruction problems via the K-theory of categories with covering families and uses this framework to reformulate the classical edge reconstruction conjecture from graph theory. For a finite graph G, the edge deck ED(G) is the multiset of its one-edge-deleted subgraphs, taken up to isomorphism. The conjecture asserts that ED(G) determines G up to isomorphism when G has at least four edges.

Within the paper’s framework, the conjecture is equivalent to injectivity of a canonical map from isomorphism classes of n-edge graphs to K0 of a tailored category with covering families (denoted Γ_{n,n−1}). Although the authors reformulate the conjecture algebraically, they do not resolve it.