The provided text introduces GraphFC, a new framework for fact-checking that converts claims into graph structures composed of subject-relation-object triplets to overcome limitations of existing language model-based methods. These methods often suffer from insufficient decomposition of claims and ambiguity in identifying mentions. GraphFC constructs claim and evidence graphs, then uses graph-guided planning to prioritize triplet verification and graph-guided checking to assess the truthfulness of these triplets by matching and completing information between the graphs. Experimental results on several datasets demonstrate that GraphFC achieves superior performance, particularly on complex claims requiring multi-hop reasoning.