This report provides the theoretical foundation to STAR program, a linear network reduction software kit. Y-$\Delta$ transformation in s-domain is the essential part of the reduction engine of the program. We have extended it to handle current/voltage sources and K elements. Coming with it are node ordering algorithm and some novel numerical stability control techniques. They play an important role in the overall reduction flow by providing the consistent accuracy in program outputs.  is dedicated to the applications to the program.
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