circuit analysis - Under what conditions is the star-mesh transform invertible?

We all know and love the Δ-Y (delta-wye) and Y-Δ (wye-delta) transforms for simplifying three-resistor networks:

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The Δ-Y and Y-Δ transforms have the nice property that a Δ can always be turned into a Y, and a Y can always be turned into a Δ, no matter the value of the resistances involved.

There's a generalised version of the Y-Δ transform called the star-mesh transform. This converts a "star" of \$ N \$ resistors into a "mesh" of \$ ^{N}C_{2} \$resistors.

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Wikipedia suggests that the star-to-mesh transform will always exist - but that the inverse transform, mesh-to-star, may not exist. To wit:

The transform replaces N resistors with \$^{N}C_{2}\$ resistors. For N > 3, the result is an increase in the number of resistors, so the transform has no general inverse without additional constraints.

What are the constraints that must be satisfied in order for the inverse to exist?

I am particularly interested in converting a 4-node mesh network into a 4-resistor star network.

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Motivation for the question: I have an industrial power systems model (really just a very large network of constant-voltage sources and impedances) containing ~ 2,000 nodes. I am attempting to reduce it to just four nodes of interest.

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Edit:

There are some published papers on this topic.

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  Versfeld, L., "Remarks on star-mesh transformation of electrical networks," Electronics Letters , vol.6, no.19, pp.597,599, September 17 1970

  
  
  

  Two new aspects of the well known star-mesh transformation are studied: (a) the necessary and sufficient conditions for the transformation of a given general mesh network into an equivalent star network; (b) an extension to networks containing sources.

  
  


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  Bapeswara Rao, V.V.; Aatre, V.K., "Mesh-star transformation," Electronics Letters , vol.10, no.6, pp.73,74, March 21 1974

  
  

  An equivalent star network exists for a given mesh network if the latter satisfies the Wheatstone relaflonship. Using this fact, it is shown that all the offdiagonal cofactors of the datum-node admittance matrix of such a mesh network are equal. From this property, a simple relationship between the elements of the two networks is derived.

  
  

I don't have IEEE Xplore access so I can't read them.