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The Sudoku puzzle has achieved worldwide popularity recently, and attractedgreat attention of the computational intelligence community. Sudoku is alwaysconsidered as Satisfiability Problem or Constraint Satisfaction Problem. Inthis paper, we propose to focus on the essential graph structure underlying theSudoku puzzle. First, we formalize Sudoku as a graph. Then a solving algorithmbased on heuristic reasoning on the graph is proposed. The related r-Reductiontheorem, inference theorem and their properties are proved, providing theformal basis for developments of Sudoku solving systems. In order to evaluatethe difficulty levels of puzzles, a quantitative measurement of the complexitylevel of Sudoku puzzles based on the graph structure and information theory isproposed. Experimental results show that all the puzzles can be solved fastusing the proposed heuristic reasoning, and that the proposed game complexitymetrics can discriminate difficulty levels of puzzles perfectly.

Heuristic Reasoning on Graph and Game Complexity of Sudoku