This paper proposes a two-step graph partitioning method to discover constrained clusters with an objective function that follows the well-known min-max clustering principle. Compared with traditional approaches, the proposed method has several advantages. Firstly, the objective function not only follows the theoretical min-max principle but also reflects certain practical requirements. Secondly, a new constraint is introduced and solved to suit more application needs while unconstrained methods can only control the number of produced clusters. Thirdly, the proposed method is general and can be used to solve other practical constraints. The experimental studies on word grouping and result visualization show very encouraging results.