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A \(2\)-connected graph is called \(Y – \Delta\) (respectively \(\Delta – Y\)) \({reducible}\) or simply a \(Y – \Delta\) (respectively \(\Delta – Y\)) graph if it can be reduced to a single edge using a sequence of \(Y – \Delta\) (respectively \(\Delta – Y\), series and parallel reductions. This paper addresses the problem of decomposing \(Y – \Delta\) and \(\Delta – Y\) graphs in connection with a new method for decomposing \(3\)-connected graphs proposed recently by Coullard, Gardner, and Wagner.