Automated fact-checking aims to assess the truthfulness of textual claims based on relevant evidence. However, verifying complex claims that require multi-hop reasoning remains a significant challenge. We propose GraphCheck, a novel framework that transforms claims into entity-relationship graphs for structured and systematic fact-checking. By explicitly modeling both explicit and latent entities and exploring multiple reasoning paths, GraphCheck enhances verification robustness. While GraphCheck excels in complex scenarios, it may be unnecessarily elaborate for simpler claims. To address this, we introduce DP-GraphCheck, a variant that employs a lightweight strategy selector to choose between direct prompting and GraphCheck adaptively. This selective mechanism improves both accuracy and efficiency by applying the appropriate level of reasoning to each claim. Experiments on the HOVER and EX-FEVER datasets demonstrate that our approach outperforms existing methods in verification accuracy, while achieving strong computational efficiency despite its multipath exploration. Moreover, the strategy selection mechanism in DP-GraphCheck generalizes well to other fact-checking pipelines, highlighting the broad applicability of our framework.