Dementia is an incurable neurodegenerative disease primarily affecting the older population, for which the World Health Organisation has set to promoting early diagnosis and timely management as one of the primary goals for dementia care. While a range of popular machine learning algorithms and their variants have been applied for dementia diagnosis, fuzzy systems, which have been known effective in dealing with uncertainty and offer to explicitly reason how a diagnosis can be inferred, sporadically appear in recent literature. Given the advantages of a fuzzy rule-based model, which could potentially result in a clinical decision support system that offers understandable rules and a transparent inference process to support dementia diagnosis, this paper proposes a novel fuzzy inference system by adapting the concept of dominant sets that arise from the study of graph theory. A peeling-off strategy is used to iteratively extract from the constructed edge-weighted graph a collection of dominant sets. Each dominant set is further converted into a parameterized fuzzy rule, which is finally optimized in a supervised adaptive network-based fuzzy inference framework. An illustrative example is provided that demonstrates the interpretable rules and the transparent reasoning process of reaching a decision. Further systematic experiments conducted on data from the Open Access Series of Imaging Studies (OASIS) repository, also validate its superior performance over alternative methods.