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The ideal (tagged resp.) triangulation of bounded surface with marked points are associated with skew-symmetric (skew-symmetrizable) exchange matrices. An algo- rithm is established to decompose the graph associated to such matrix. There are finite many graph with non-unique decomposition. We find all such graphs and their decompositions. In addition, we also find the associated ideal (tagged) triangulations to different decompositions.arXiv:1112.1008v1 fatcat:rxp4oih5sjenrprgah6ug5hmoe