On the minimum hybrid rank of a graph relative to a partition of its edges and its application to electrical network analysis

Abstract

AbstractWe consider the following problem in this paper: given a partition of the set of edges of a graph such that the subgraphs on the blocks of the partition are connected, what is the minimum number of node fission (splitting a node into two) and node fusion (coalescing two nodes) operations that will destroy all circuits containing edges from more than one block of the partition? This problem can be seen to be a generalization of the problem of determining the minimum hybrid rank of a graph. We may therefore regard this minimum number as the minimum hybrid rank of the graph relative to a specified partition of the edge set of the graph. We give an efficient algorithm for the construction of a minimum length sequence of such operations. Our solution can be applied to the problem of analysing networks after first decomposing them into subnetworks‐the equations of the network can be displayed in a bordered block diagonal form where the blocks along the diagonal correspond to the subnetworks and the thickness of the border corresponds to the minimum hybrid rank relative to the partition of the sets of edges of the network into the sets of edges of the subnetworks.