Do LLMs Brainstorm Better with Knowledge Graphs?

In AI-assisted product ideation, a knowledge graph is an attractive way to represent the solution space for LLM-powered combinatorial search, yet few studies ask how the form of the graph given to the model affects what it produces, and at what cost. We test this on a microwave oven described as a graph (123 nodes, 246 edges) linking user needs to functions, parts and technologies. For each of 13 needs, Claude Haiku got the whole graph, the slice relevant to that need, a random slice of the same size, or no graph, at four effort levels; Claude Sonnet scored the 2,892 ideas blind, and we counted top ideas (roughly the best 12%) per 100k tokens. A follow-up asked Haiku, Opus and Fable for 50 ideas per call on one need at medium effort (6,091 ideas). The full graph never paid off. It yielded about 3 top ideas per 100k tokens at every effort, against 8.8 for the slice at medium effort; at high effort, the only one at which all four conditions ran, the slice yielded 5.9 and no graph 6.4. In the follow-up, on a need chosen to match this ordering, the full graph was the least efficient for every model, clearly so for Haiku and Fable; for the stronger models it added cost without lowering the share of good ideas. The slice was the best way to give the graph but no better than no graph. The planned comparison of whole idea sets favoured the full graph (partly reflecting length); the yield measure was defined after seeing the results and is exploratory. Stronger models found 2 to 2.5 times as many top ideas per token, yet Haiku found 10 to 40 times more per dollar, an advantage that reverses when people, not an automated judge, screen the ideas. Thinking longer raised quality, not efficiency. An informal reading of all first-study ideas by one author, without a p