A Proof of the Graph Reconstruction Conjecture: Residual Arrows, Double Deletion, Equal Heaps, and Global Zero Residue

AbstractWe present a purely geometric residual-arrow construction for graph reconstruction. The starting data are the completely known unlabeled (n−1)(n−1)-vertex cards. Inside each card, every undirected edge is rewritten as two opposite arrows. A second vertex deletion cuts these paired arrows and leaves outward arrows at their surviving endpoints. The arrow carries no label and does not record the identity of the deleted endpoint; it is only a geometric outward trace. Consequently every resulting (n−2)(n−2)-vertex fragment carries both an internal graph structure and a boundary-arrow pattern. Since every two (n−1)(n−1)-vertex cards necessarily overlap in an (n−2)(n−2)-vertex fragment, the complete second-deletion pool admits an equal division into nn heaps of n−1n−1 arrowed fragments. Some heaps may be identical, but heap identity is not used as the final discriminator. The heaps are instead glued by pairing opposite arrows, and only a gluing with zero global residue is accepted. The paired overlaps read every adjacency entry by comparing the known deleted-vertex degree with the number of residual arrows. In the nonregular branch, degree loads provide anchors and the complete arrow profiles distinguish all nonautomorphic placements. Therefore every wrong pairing leaves a residual arrow, while every residue-free exchange is an automorphism. The resulting reconstruction is unique up to isomorphism. Regular graphs, including complete and empty graphs, are handled separately. KeywordsGraph reconstruction conjecture; residual arrows; double deletion; equal heaps; global zero residue; vertex-deleted subgraphs; geometric proof; automorphism; regular graphs; strongly regular graphs.