Proof Graph Reconstruction

• Proof graph reconstruction is the process of recovering complete logical or computational proof structures from partial subgraph information using graph-theoretic and algebraic methods.
• It leverages modular decomposition and symmetry properties to identify core logical dependencies and reconstruct underlying structures from vertex or edge deletions.
• Algorithmic approaches often yield polynomial-time methods, supporting practical applications in formal verification and enhancements in graph neural network models.

Proof graph reconstruction is the process of recovering the structure of a proof graph—typically a representation of a logical argument, inference structure, or computational workflow—given only partial or local information such as its collection of vertex- or edge-deleted subgraphs, smaller induced subgraphs, or certain combinatorial invariants. The concept draws from classical graph reconstruction theory, especially the Ulam-Kelly reconstruction conjecture, but often targets structured graph classes (e.g., modular, interval, tree-like, or bounded-degree graphs) or focuses on “proof graphs” arising in computer science, mathematics, and logic. Proof graph reconstruction techniques exploit unique decompositions, symmetry properties, algebraic constraints, or combinatorial structures to guarantee that the entire proof structure can be consistently recovered from parts.

1. Foundations: Classical Reconstruction and Modular Decomposition

The Ulam-Kelly reconstruction conjecture forms the cornerstone of graph reconstruction. It posits that any finite simple graph with at least three vertices is uniquely determined, up to isomorphism, by its multiset of vertex-deleted subgraphs (the deck). For a proof graph G representing inference steps or dependencies, this translates to reconstructing G from partial “views” (subproofs) where specific statements or inferences are removed.

A major methodological advance is modular decomposition, wherein G is canonically factored as the “inflation” of an indecomposable skeleton K, with each vertex k ∈ K replaced by a module (interval) Iₖ. Formally, G = K[Iₖ : k ∈ K], where intervals have the property that their external neighborhoods are identical for all u, v ∈ Iₖ:

$N(u) \setminus V(Iₖ) = N(v) \setminus V(Iₖ)$

This decomposition is unique for |K| > 2. Reconstruction proceeds by identifying both the skeleton and the intervals from the deck, using combinatorial and group-theoretic properties (distinguishing cases with multiple, single, or small-size nonsingleton intervals) (Brignall et al., 2011). When applied to proof graphs, the skeleton encodes the irreducible core logical dependencies, and intervals represent modular clusters of related statements.

2. Specialized Graph Classes and Structure Theorems

Many classes of graphs arising in proof formalism or algorithm design admit efficient and unique reconstruction schemes:

• Interval graphs: These are graphs whose vertices correspond to intervals on a line, with edges indicating overlapping intervals. Using structural parameters (bulk, flanks, outsiders) and decomposing along “clean clique separators,” it is possible to reconstruct an interval graph from its deck in polynomial time, provided careful handling of separations and resilience under vertex deletion (Heinrich et al., 3 Apr 2025). Separators are characterized via tidy interval representations ensuring the separator’s overlap is strictly between A and B (no edges between these outside classes), i.e.,

$$\bigcap_{c\in C} [\ell_c, r_c] \subset \left(\max_{a \in A} r_a,\; \min_{b \in B} \ell_b \right)$$

This methodology is widely applicable in reconstructing interval-structured proof graphs.

• Graphs as cubes of trees: Graphs isomorphic to the third power of a tree can be reconstructed by leveraging the deck’s detailed clique and peripheral structures. The result hinges on the recognizability of the cube class, the uniqueness of the square/cube root (except in degenerate complete cases), and leveraging known reconstructibility of trees from peripheral-deleted subgraphs (Gupta et al., 2012).
• Triangle-free and bounded-degree graphs: For triangle-free graphs, connectivity queries on all k-vertex subsets suffice for unique reconstruction (for k ≥ 3 and sufficiently large graphs), and for bounded-degree graphs, all consistent structures share a “skeleton,” differing only locallly (Kluk et al., 10 Jul 2024).
• 3-regular graphs: These are shown to be 2-reconstructible; the deck of (n–2)-vertex induced subgraphs suffices to recover the unique original graph up to isomorphism, relying on the interaction of short cycles and the structure of their overlaps (Kostochka et al., 2019).

3. Algebraic and Combinatorial Frameworks

Algebraic formalism plays a central role in understanding and quantifying reconstructibility. Kocay’s Lemma associates to each deck and sequence of graphs a linear constraint on subgraph counts, summarized in the formula

$$\sum_{H} c(\mathcal{F}, H) \cdot s(H, G) = k_{\mathcal{F},R}$$

where $c(\mathcal{F}, H)$ counts covers of H by the sequence, and $s(H, G)$ counts subgraphs isomorphic to H in G. Organizing such constraints into a matrix $M_{\mathcal{F}, \mathcal{C}_n}$, the rank of this matrix lower-bounds the number of distinct decks (reconstruction classes):

$d(n) \geq \mathrm{rank}_\mathbb{R}(M)$

If the rank matches the number of graph isomorphism classes, reconstructibility holds. Moreover, these algebraic perspectives generalize to directed graphs, hypergraphs, and restricted subgraph families, allowing precise characterization of when reconstruction is possible or, conversely, why counterexamples may exist (Oliveira et al., 2013).

4. Role of Asymmetry, Anchors, and Local Uniqueness

Structural asymmetry, the presence of “anchors,” and unique substructures underpin many concrete proofs of reconstructibility.

• Anchors: An anchor is a proper induced subgraph H that appears exactly once in G. This substructure allows H to be consistently identified in every card containing it, permitting a reduction of the reconstruction problem to that of a simpler “shadow graph” derived from H. Connectional anchors extend this to subgraphs made unique via their pattern of attachment to the remainder of the graph. For many non-regular graphs, such unique or nearly unique subgraphs almost always exist, and extending them to maximal anchors can reduce the problem to reconstructing highly symmetric residues (Farhadian, 2017, Farhadian, 2016).
• Shadow graphs: After identifying an anchor, the relationships and attachments of the residual vertices to the anchor are encoded in a shadow graph, whose reconstructibility often follows from structural or symmetry properties.
• Automorphism group and orbits: In graphs lacking high symmetry (i.e., not vertex-transitive), orbits can be “pinned” by removal, and unique reconstruction follows by combining anchor information and automorphism group constraints.

5. Extensions: Edge Reconstruction, Metric Graphs, and Noisy Data

Advanced forms of reconstruction address alternative information sources and practical constraints:

• Edge reconstruction: In the edge-deleted deck model, the recovery of the graph is tied to spectral invariants and walk structures, such as the Ihara zeta function—reconstructible for graphs with average degree ≥4 (Cornelissen et al., 2015). Measure-theoretic rigidity shows that for graphs of large enough first Betti number, knowledge of the measure on the boundary of the universal covering tree (as encoded combinatorially by lengths and overlaps of closed walks) suffices for complete reconstruction (Cornelissen et al., 2016).
• Metric and embedded graphs under noise: For graph-like structures arising from sampled geometric or topological data (e.g., point clouds near an embedded graph), neighborhood (Vietoris–Rips) graphs combined with α-Reeb graph constructions and (ε, R)-approximations produce robust geometric reconstructions under relaxed density and Hausdorff noise conditions. Theoretical error bounds tie the structure of the reconstructed graph to the ground truth via parameters such as Betti number, edge lengths, and restricted distortion (Fritze et al., 25 Oct 2024).

6. Algorithmic and Practical Implications

Graph and proof graph reconstruction results have significant algorithmic consequences:

• Polynomial-time reconstruction: For interval graphs, the developed techniques yield polynomial-time algorithms to reconstruct the graph from its deck (Heinrich et al., 3 Apr 2025). Path graph reconstruction, as for non-crossing spanning paths on convex point sets, admits an $O(N \log N)$ algorithm using degree signatures and clique analysis (Keller et al., 2017).
• Distributed and query-based reconstruction: In distributed models (e.g., the congested clique), information-theoretic lower bounds match the bandwidth complexity of optimal algorithms that can reconstruct a global structure from only local views and communication, leveraging graph fingerprints and error-correcting codes (Montealegre et al., 2017). In query models, enumeration algorithms reconstruct all consistent graphs given k-set connectivity, with unique reconstruction for triangle-free and bounded-degree cases (Kluk et al., 10 Jul 2024).
• Implications for Graph Neural Networks: Proof graph reconstruction techniques inform graph representation learning. Approaches that aggregate representations over induced subgraphs of varying sizes (the k-deck) boost the expressive power of GNNs, enable the discovery of hereditary invariances, and enhance empirical performance beyond classical message-passing architectures (Cotta et al., 2021).

7. Limitations and Open Problems

Technical limitations persist in characterizing which graph classes admit unique and efficient reconstructibility, especially for highly symmetric, regular, or complex-structured graphs. Certain problems become NP-hard or information-theoretically intractable when only partial local connectivity information is supplied (e.g., partial k-query models for k ≥ 4) (Kluk et al., 10 Jul 2024). For more general abstract settings, identifying minimal obstructions, tight lower bounds, or unique invariants remains challenging.

Key open questions include:

• Extending polynomial-time reconstruction to broader classes beyond interval, modular, and triangle-free graphs.
• Understanding and relaxing noise/distortion assumptions for geometric and metric proof graph reconstruction.
• Unifying algebraic, combinatorial, and measure-theoretic approaches for application to directed, weighted, or colored proof graphs.
• Determining the minimal sufficient local information for unique reconstruction in practical verification and AI systems.

These unresolved directions continue to stimulate advances in both graph theory and its applications to formal reasoning, verification, and data analysis.