GraphFC: Graph-Based Decomposition

• Graph-based Decomposition (GraphFC) is a modular framework that decomposes claims into atomic triplets, enhancing clarity through structured evidence graphs.
• It employs graph-guided planning and matching to prioritize fully-grounded triplets, reducing ambiguity and boosting multi-hop verification precision.
• Empirical results show significant gains over traditional methods on benchmarks like HOVER, FEVEROUS, and SciFact, demonstrating its state-of-the-art performance.

A graph-based decomposition framework, typified by GraphFC, represents a principled approach to fact verification that explicitly encodes both claims and supporting evidence as structured graphs of atomic triplets, and orchestrates their verification via graph-theoretic planning and matching procedures. This paradigm addresses key limitations of conventional (often purely text-based) claim decomposition, namely insufficient atomicity and pervasive referential ambiguity, by leveraging relational and coreference constraints inherent in graph structures (Huang et al., 10 Mar 2025). The result is a modular, interpretable, and empirically robust pipeline for fine-grained, multi-hop fact-checking.

1. Formalism: Claim and Evidence Graph Construction

Let $C$ be a natural-language claim and $E$ the associated evidence text or corpus. A graph-based decomposition begins by parsing $C$ into a claim graph $G_c$ composed of directed triplets $t=(s,p,o)$, where:

• $s \in \mathcal{E}_c \cup \mathcal{X}_c$: a known entity (named in $C$) or an unknown placeholder (to be grounded later).
• $p \in \mathcal{R}_c$: a relation/verb phrase from $C$.
• $o \in \mathcal{E}_c \cup \mathcal{X}_c$: analogous for the object.

Formally,

$$G_c = (\mathcal{V}_c,\,\mathcal{E}_c), \quad \mathcal{V}_c = \{ t_i^c : i = 1, \ldots, n_c \}, \quad \mathcal{E}_c = \left\{ (t_i^c, t_j^c) : \exists x \in \mathcal{E}_c \cup \mathcal{X}_c, x \in \{s_i,o_i\} \cap \{s_j,o_j\} \right\}$$

Edges express referential overlap, enabling propagation of grounded placeholders across triplets.

Similarly, the evidence graph $G_e$ is constructed from $E$ by extracting triplets $t_j^e = (s_j^e, p_j^e, o_j^e)$ involving known entities in $\mathcal{E}_c$ but devoid of unknowns. These graphs are derived using LLM-based information extraction or standard NLP toolkits, encapsulated here as agents $\mathcal{A}_{gc}$ ($f_c$ for claim, $f_e$ for evidence).

2. Graph-Guided Planning: Verification Sequencing

Given $G_c$, the next phase involves a graph-guided planner $\mathcal{S}_{gp}$, which determines the optimal order in which to attempt verification of claim triplets. Each triplet is assigned a priority $\rho(t)$, based on the number of unresolved entities: $$\rho(t) = \begin{cases} 0 & \text{if } s, o \in \mathcal{E}_c \ 1 & \text{if exactly one of } s, o \in \mathcal{X}_c \ 2 & \text{if } s, o \in \mathcal{X}_c \end{cases}$$ The planner sorts triplets by ascending $\rho(t)$, ensuring that those with fully-grounded entities (atomic and directly verifiable) are acted on first. The output is an ordered list $\mathcal{T} = [\hat{t}_1^c, ..., \hat{t}_{n_c}^c]$.

3. Graph-Guided Checking: Triplet-wise Verification

For each $\hat{t}_i^c$ (per planner order), the system executes one of two verification procedures:

• Graph Match ($\rho=0$): Performs direct subgraph matching between $\hat{t}_i^c$ and candidate triplets in $G_e$ sharing both $s$ and $o$. If an exact match is found, $Y_{\hat{t}} = \text{True}$; otherwise $\text{False}$.
• Graph Completion ($\rho=1$): For triplets with an unknown entity (e.g., $s \in \mathcal{X}_c$), searches $G_e$ to propose a concrete grounding $e \in \mathcal{E}_c$, yielding $(e, Y_{\hat{t}})$; $Y_{\hat{t}}=\text{True}$ if grounding is possible, else $\text{False}$. Successive claim graph triplets that reference this entity are updated with the grounded value.

The claim is supported if and only if all triplets verify as $$Y = \bigwedge_{i=1}^{n_c} Y_{\hat{t}_i^c} = \text{True}$$.

4. Decomposition Granularity and Coreference Resolution

GraphFC ensures that each claim is decomposed into minimal atomic triplets—no under-decomposition (insufficient splitting) occurs, as each $(s,p,o)$ asserts a single, irreducible fact, and no over-decomposition (spurious splits) arises, because triplets are preserved as logical wholes. Referential ambiguity is actively resolved: placeholders $\mathcal{X}_c$ connect subclaims, and entity groundings are consistently propagated, avoiding contradictory interpretations of co-referring expressions.

Empirical analysis demonstrates that the resulting graphs typically contain twice as many triplets per claim as generic LLM-based decompositions, but this increased granularity correlates with improved factual precision and recall (Huang et al., 10 Mar 2025).

5. Empirical Performance and Ablation Insights

Extensive benchmarking on multi-hop verification datasets shows that GraphFC achieves state-of-the-art macro-F1 in both gold-evidence and open-domain settings. On HOVER (4-hop, open), GraphFC surpasses the best prior method (ProgramFC) by 8.31 points (67.47% vs. 59.16% macro-F1). Comparable gains are observed on FEVEROUS (+5.08 points, 72.88%) and SciFact (+7.71 points, 80.63%).

Ablation studies reveal:

Component	F1 Impact (4-hop HOVER, open)
Remove evidence graph	–6 points
Remove graph-guided plan	–7 points
Replace graph construction LLM	Minor decrease (graph match/completion robust)

Both evidence graph construction and graph-guided planning are crucial for multi-hop and cross-referential claims; their removal substantially degrades accuracy.

6. Context and Comparative Advances

Graph-based decomposition directly overcomes the key failings of conventional claim segmentation—namely, insufficient decomposition (failure to separate intertwined facts) and referential ambiguity (failure to propagate grounded entities across sub-claims). By structuring the verification process as iterative triplet grounding and relational reasoning, it delivers both theoretical clarity (minimal, unambiguous atomic units) and practical gains (fine-grained verification, transparency, and error propagation mitigation).

This establishes GraphFC as a benchmark for structured reasoning in fact-checking, aligning atomicity, verifiability, and referential fidelity, and supporting robust, multi-hop claim verification pipelines in open-domain and specialized settings (Huang et al., 10 Mar 2025).