Anyone who's still pursuing this topic might be interested in looking at the
engineers' approach to it. As a student (almost 40 years ago) I took a
sequence of circuit courses in the EE department, and I recall that the text
had a whole chapter on "network topology" devoted to exactly this question
of most efficiently choosing a system of independent equations for solving a
circuit. The underlying ideas are no doubt the same ones included in the
discussion here, but the format is a little different (as is often the case
in engineering texts). The book was Wallace Cassell's Linear Electric
Circuits (Wiley, 1964), and I suppose most similar engineering texts would
have such a chapter.