Knowledge Graph Question Answering (KGQA) has advanced through structured query generation, yet most efforts target RDF/SPARQL, leaving Cypher and property graphs underexplored, despite increasing demand for unified KGQA in industry settings. We propose UniQGen, a novel constraint-based framework that employs LLM agents to dynamically extract and refine representative graph query clauses into executable, intent-aligned graph queries across query languages. The foundation of our method is a variant of Chase & Backchase, a family of algorithms for query optimization and reformulation. We extend Chase & Backchase with a dynamic reasoning process over query constraints that also interact with LLMs for query quality estimation. With a Cypher-supported Freebase graph deployed on Amazon Neptune, we extensively evaluate our approach on popular KGQA benchmarks (GraphQ, GrailQA, and WebQSP). We demonstrate that UniQGen outperforms state-of-the-art graph query generation techniques in both accuracy and efficiency, with F1 gains of 31.6% on GraphQ and 4.9% on GrailQA. Unlike prior methods, our framework does not require fine-tuning for schema matching, making it more extensible to schema-less graphs and semantics in query workloads, and is more suitable for enterprise-grade KGQA. We release Cypher outputs and a Neptune-ready Freebase snapshot to support reproducible, cross-language KGQA research.

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