Can delta-star conversions be used to resolve any circuit of resistors into series and parallel combinations of resistors? I mean to say that if we do not utilize any other techniques like merging points with the same potential and so on, can we reduce any circuit to a simple one where all resistances are in some combination of parallel and series connections?
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2$\begingroup$ Hint: think about non-planar circuits. $\endgroup$– The PhotonCommented Apr 5, 2017 at 18:52
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From Wikipedia: Every two-terminal network represented by a planar graph can be reduced to a single equivalent resistor by a sequence of series, parallel, $Y$-$\Delta$, and $\Delta$-$Y$ transformations. However, there are non-planar networks that cannot be simplified using these transformations, such as a regular square grid wrapped around a torus, or any member of the Petersen family.
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1$\begingroup$ Somebody: "does every graph ..." - Petersen graph: "No." $\endgroup$ Commented Jan 23, 2022 at 19:05