As large language models (LLMs) become increasingly prevalent, ensuring their robustness against adversarial misuse is crucial. This paper introduces the GAP (Graph of Attacks with Pruning) framework, an advanced approach for generating stealthy jailbreak prompts to evaluate and enhance LLM safeguards. GAP addresses limitations in existing tree-based LLM jailbreak methods by implementing an interconnected graph structure that enables knowledge sharing across attack paths. Our experimental evaluation demonstrates GAP's superiority over existing techniques, achieving a 20.8% increase in attack success rates while reducing query costs by 62.7%. GAP consistently outperforms state-of-the-art methods for attacking both open and closed LLMs, with attack success rates of >96%. Additionally, we present specialized variants like GAP-Auto for automated seed generation and GAP-VLM for multimodal attacks. GAP-generated prompts prove highly effective in improving content moderation systems, increasing true positive detection rates by 108.5% and accuracy by 183.6% when used for fine-tuning. Our implementation is available at https://github.com/dsbuddy/GAP-LLM-Safety.