  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.
