A circuit characterization of principal and refined partition of graphs

The principal partition and the refined partition of a graph are useful in hybrid analysis of electrical networks, and in the solutions of some noncircuit problems. Definitions of such partitions of a graph are normally based on concepts from graph theory. The purpose is to provide alternative definitions of these concepts without the use of any graph-theoretic concepts such as maximally distant tree pair, extremal tree, hybrid tree, or complementary tree. It is shown that even the concept of tree is not essential to the intrinsic properties of the principal and refined partition of a graph. The author shows that the partitioning can be defined strictly using basic concepts in traditional circuit analysis, such as nodal analysis