Abstract: Prompt optimization for text-to-image (T2I) generation has been pursued almost entirely as text rewriting, in which a short user brief is expanded into a longer, model-preferred token sequence. We argue that such a language-space formulation is ill-suited to structured visual design tasks such as logo creation, where a one-line brief leaves most design decisions unspecified. These decisions depend on relational priors that a linear sequence cannot encode, and they leave an uncontrolled channel through which protected marks may be reproduced. We therefore recast logo prompting as sampling within a structured design space, and instantiate this idea as DOGS (Design-space prompting with an Originality-aware GFlowNet Sampler). From a large corpus of real-world logos, we mine a typed, graph-structured design grammar whose edges record empirical co-occurrence. A GFlowNet sampler then generates design graphs with probability proportional to a terminal reward that combines recognizability, aesthetics, and corpus-relative originality. Every slot draws only from a closed design-level vocabulary, and any infringement-inducing or harmful token is removed during parsing. The originality reward further penalizes proximity to existing logos, thereby incorporating infringement avoidance into the method by construction. On two open-source renderers and against nine baselines, DOGS produces logos that are more recognizable and aesthetic, substantially more diverse, and far less prone to trademark infringement.