Graph Reconstruction through Rooted Algebras and Card Symmetries: Nonrigid reconstruction and bounded parent lists

An ordinary graph deck records every vertex-deleted graph but forgets how the cards fit together. This working paper studies how algebra, symmetry, and compatible overlaps recover that missing information. The principal theorems reconstruct every nonrigid and every decomposable permutation graph of order at least three. Further results settle specified rigid prime classes through pendant paths, indecomposability structure, and symmetric intersections among four or five prime cards. Every ordinary permutation-graph deck of order at least six has at most eight nonisomorphic parents. Two ideas organize the arguments: rooted counts recover vertices as atoms of an algebra, while permutation diagrams restrict possible card alignments. Signed shortest paths resolve coherent gluing ambiguities under explicit sign hypotheses. For arbitrary graphs, matching card occurrences excludes rank-three adjacency differences and confines rank-two differences to cyclic spaces with disjoint coordinate supports. At walk corank three, at most four parents remain. The exposition emphasizes analytical proofs and explanatory vector figures, with exact computational certificates separated from the general arguments. Full permutation-graph reconstruction remains open. The remaining questions concern compatibility across incomplete intersections, reconstruction of bounded exterior attachments, and separation of the surviving algebraic reflections by ordinary cards.