# Cut-Set Matrix and Node-Pair Potential

**A cut-set is a minimum set of elements that when cut, or removed, separates the graph into two groups of nodes. A cut-set is a**

*Cut-Set***minimum set of branches**of a connected graph, such that the removal of these branches from the

**graph reduces the rank of the graph by one**.

In other words, for a given connected graph (G), a set of branches (C) is defined as a cut-set if and only if:

- the removal of all the branches of C results in an unconnected graph.
- the removal of all but one of the branches of C leaves the graph still connected.

# Fundamental Cut-Set

A fundamental cut-set (FCS) is a cut-set that cuts or contains**one and only one tree branch**. Therefore, for a given tree, the number of fundamental cut-sets will be equal to the number of twigs.

# Properties of Cut-Set

- A cut-set divides the set of nodes into two subsets.
- Each fundamental cut-set contains one tree-branch, the remaining elements being links.
- Each branch of the cut-set has one of its terminals incident at a node in one subset and its other terminal at a node in the other subset.
- A cut-set is oriented by selecting an orientation from one of the two parts to the other. Generally, the direction of cut-set is chosen same as the direction of the tree branch.

# Cut-Set Matrix (*Q*_{C})

For a given graph, a cut-set matrix (_{C}

*Q*) is defined as a rectangular matrix whose rows correspond to cut-sets and columns correspond to the branches of the graph. Its elements have the following values:

_{C}*Q*= 1, if branch

_{ij}*j*is in the cut-set

*i*and the orientations coincide.

= â€“1, if branch

= 0, if branch

*j*is in the cut-set*i*and the orientations do not coincide.= 0, if branch

*j*is not in the cut-set*i*.# Cut-Set Matrix and KVL

By cut-set schedule, the branch voltages can be expressed in terms of the tree-branch voltages.A cut-set consists of

*one and only one*branch of the tree together with any links which must be cut to divide the network into two parts. A set of fundamental cut-sets includes those cut-sets which are obtained by applying cut-set division for each of the branches of the network tree.

**(a) Graph**

**(b) Tree**

**To summarize**, KVL and KCL equations in three matrix forms are given below.

Matrix |
KCL |
KVL |

Incidence Matrix (A)_{a} |
A Ã— _{a}I = 0_{b} |
V = _{b}A_{a} Ã— ^{T}V_{n} |

Tie-Set Matrix (B)_{a} |
I = _{b}B_{a} Ã— ^{T}I_{L} |
B Ã— _{a}V = 0_{b} |

Cut-Set Matrix (Q)_{C} |
Q Ã— _{C}I = 0_{b} |
V = _{b}Q_{C} Ã— ^{T}V_{t} |