When we talk of cut set matrix in graph theory, we generally talk of fundamental cut-set matrix. A cut-set is a minimum set of branches of a connected graph such that when removed these branches from the graph, then the graph gets separated into 2 distinct parts called sub-graphs and the cut set matrix is the matrix which is obtained by row-wise taking one cut-set at a time. The cutset matrix is denoted by symbol [Qf].
Example of Cutsets Matrix of a Circuit
Two sub-graphs are obtained from a graph by selecting cut-sets consisting of branches [1, 2, 5, 6].
Thus, in other words we can say that fundamental cut set of a given graph with reference to a tree is a cut-set formed with one twig and remaining links. Twigs are the branches of tree and links are the branches of co-tree.
Thus, the number of cutset is equal to the number of twigs.
[Number of twigs = N – 1]
Where, N is the number of nodes of the given graph or drawn tree.
The orientation of cut-set is the same as that of twig and that is taken positive.
Steps to Draw Cut Set Matrix
There are some steps one should follow while drawing the cut-set matrix. The steps are as follows-
- Draw the graph of given network or circuit (if given).
- Then draw its tree. The branches of the tree will be twig.
- Then draw the remaining branches of the graph by dotted line. These branches will be links.
- Each branch or twig of tree will form an independent cut-set.
- Write the matrix with rows as cut-set and column as branches.
| Branchase ⇒ | 1 | 2 | 3 | . | . | b | |
| Cutsets | |||||||
| C1 | |||||||
| C2 | |||||||
| C3 | |||||||
| . | |||||||
| . | |||||||
| Cn | |||||||
n = number of cut-set.
b = number of branches.
Orientation in Cut Set Matrix
Qij = 1; if branch J is in cut-set with orientation same as that of tree branch.
Qij = -1; if branch J is in cut-set with orientation opposite to that of branch of tree.
Qij = 0; if branch J is not in cut-set.
Example 1
Draw the cut-set matrix for the following graph.
Answer:
Step 1: Draw the tree for the following graph.
Step 2: Now identify the cut-set. Cut-set will be that node which will contain only one twig and any number of links.
Here C2, C3 and C4 are cut-sets.
Step 3: Now draw the matrix.
| Branchase ⇒ | 1 | 2 | 3 | 4 | 5 | 6 | |
| Cutsets | |||||||
| C2 | +1 | +1 | 0 | 0 | -1 | 0 | |
| C3 | 0 | 0 | +1 | 0 | +1 | -1 | |
| C4 | -1 | 0 | 0 | +1 | 0 | +1 | |
This is the required matrix.
Example 2:
Draw the cut-set of the given graph.
Answer:
Again in this question we have to repeat the same steps as done in previous question.
Step 1: Draw the tree for the following graph.
Step 2: Now identify the cut-set. Cut-set will be that node which will contain only one twig and any number of links.
Here C1 and C5 are cut-sets.
Step 3: Now draw the matrix.
| Branchase ⇒ | 1 | 2 | 3 | 4 | 5 | |
| Cutsets | ||||||
| C1 | +1 | +1 | -1 | -1 | 0 | |
| C5 | 0 | -1 | 0 | -1 | +1 | |
This is the required matrix.
Points to remember
There are some key points which should be remembered. They are:-
- In cutset matrix, the orientation of twig is taken positive.
- Each cut-set contains only one twig.
- Cut-set can have any number of links attached to it.
- The relation between cut-set matrix and KCL is that
