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LLM-based knowledge-graph question answering (KGQA) delegates graph traversal to language models, turning each question into a sequence of local relation-selection decisions repeated across beams and hops. A common but untested default is to serialize the complete partial path into every routing prompt, even though the controller already maintains this path as exact symbolic state. Bounded Path Context (BPC) decouples these two roles: the controller retains full paths in symbolic memory for answer extraction and audit, while the relation-selection prompt exposes only the question, the current entity, outgoing relation candidates, and at most the last K hops. A controlled sweep over K -- fixing graph neighborhoods, beam budget, depth, decoding, and answer-extraction format -- shows that bounded histories match or exceed full-history prompting on complete WebQSP and CWQ test sets with Qwen3.5-9B-AWQ: K=1 achieves 0.487 answer-set F1 on WebQSP versus 0.472 for full history, and K=0 reaches 0.287 on CWQ versus 0.274, with 9.7% and 12.1% fewer input tokens respectively. At the 4B scale, K=1 remains the strongest setting on both benchmarks. Per-example analysis reveals that 71-84% of examples are unaffected by history length, while the affected cases expose when prior hops disambiguate versus distract. These results suggest that path serialization length is better treated as a tunable interface variable than as a default assumption in LLM-based graph controllers.