- Cut-Set Matrix Definition: A cut-set matrix is defined as a matrix showing the cut-sets of a connected graph in an electric circuit.
- Cut-Set Concept: A cut-set is the minimum set of branches that separates a graph into two sub-graphs when removed.
- Fundamental Cut-Set: Formed using one twig and the remaining links, fundamental to understanding graph structures.
- Matrix Construction: To construct the cut-set matrix, identify twigs and links, and form rows for cut-sets and columns for branches.
- Orientation in Matrix: Orientation of branches in the cut-set matrix is positive if it matches the twig’s orientation, negative if opposite, and zero if not included.
In cut-set matrix in graph theory, a cut-set is a minimal set of branches whose removal disconnects a connected graph. A fundamental cut-set matrix is built after choosing a spanning tree. Each row represents the fundamental cut-set associated with one tree branch, or twig. The matrix is denoted by [Qf].
Example of Cutsets Matrix of a Circuit
Removing the cut-set consisting of branches [1, 2, 5, 6] divides the shown connected graph into two components.
For a chosen tree, remove one twig to split the tree into two parts. That twig and every link crossing between those parts form one fundamental cut-set. Twigs belong to the tree, while links are the remaining graph branches.
The number of fundamental cut-sets equals the number of twigs in the selected tree.
[Number of twigs = N – 1]
Here N is the number of nodes in a connected graph.
The chosen orientation of each fundamental cut-set is normally the orientation of its defining twig, which gives that twig a positive entry.
Steps to Draw Cut Set Matrix
Use the following steps to construct the cut-set matrix for a connected, directed graph:
- Draw the directed graph of the network and label every branch.
- Select a spanning tree. Its N – 1 branches are the twigs.
- Identify the remaining branches as links. A dotted style can distinguish them in the drawing.
- Remove each twig in turn. Record that twig and every link connecting the two resulting tree components as one fundamental cut-set.
- Write one row per fundamental cut-set and one column per graph branch, using a consistent branch order.
| Branches ⇒ | 1 | 2 | 3 | . | . | b | |
| Cut-sets | |||||||
| C1 | |||||||
| C2 | |||||||
| C3 | |||||||
| . | |||||||
| . | |||||||
| Cn | |||||||
n = number of fundamental cut-sets, equal to N – 1 for a connected graph.
b = total number of graph branches.
Orientation in Cut Set Matrix
Qij = +1 if branch j belongs to cut-set i and points in the selected cut-set direction.
Qij = -1 if branch j belongs to cut-set i and points opposite to that direction.
Qij = 0 if branch j does not belong to cut-set i.
Example 1
Construct the fundamental cut-set matrix for the directed graph.
Answer:
Step 1: Select the tree shown for the graph.
Step 2: Remove each twig in turn and include every link that crosses between the two resulting tree components.
The diagram labels the resulting fundamental cut-sets C2, C3 and C4.
Step 3: Enter each branch orientation in the corresponding matrix row.
| Branches ⇒ | 1 | 2 | 3 | 4 | 5 | 6 | |
| Cut-sets | |||||||
| C2 | +1 | +1 | 0 | 0 | -1 | 0 | |
| C3 | 0 | 0 | +1 | 0 | +1 | -1 | |
| C4 | -1 | 0 | 0 | +1 | 0 | +1 | |
The table is the matrix for the branch directions shown.
Example 2:
Construct the fundamental cut-sets of the given graph.
Answer:
Use the same tree-partition method and retain the stated branch directions.
Step 1: Select the tree shown for the graph.
Step 2: Remove each twig in turn and include every link crossing the resulting partition.
The diagram labels the fundamental cut-sets C1 and C5.
Step 3: Enter the signed branch memberships in the matrix.
| Branches ⇒ | 1 | 2 | 3 | 4 | 5 | |
| Cut-sets | ||||||
| C1 | +1 | +1 | -1 | -1 | 0 | |
| C5 | 0 | -1 | 0 | -1 | +1 | |
The table is the matrix for the branch directions shown.
Points to remember
Apply these definitions consistently:
- In a fundamental cutset matrix, orient each row with its defining twig so the twig entry is positive.
- Each fundamental cut-set contains exactly one twig from the selected tree.
- A fundamental cut-set also contains every link crossing the partition made by deleting that twig.
- With the same branch-current references, the cut-set form of KCL is:
