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 [Q_{f}].

## 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 |
|||||||

C_{1} |
|||||||

C_{2} |
|||||||

C_{3} |
|||||||

. | |||||||

. | |||||||

C_{n} |

n = number of cut-set.

b = number of branches.

## Orientation in Cut Set Matrix

Q_{ij} = 1; if branch J is in cut-set with orientation same as that of tree branch.

Q_{ij} = -1; if branch J is in cut-set with orientation opposite to that of branch of tree.

Q_{ij} = 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 C_{2}, C_{3} and C_{4} are cut-sets.

Step 3: Now draw the matrix.

Branchase ⇒ | 1 | 2 | 3 | 4 | 5 | 6 | |

Cutsets |
|||||||

C_{2} |
+1 | +1 | 0 | 0 | -1 | 0 | |

C_{3} |
0 | 0 | +1 | 0 | +1 | -1 | |

C_{4} |
-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 C_{1} and C_{5} are cut-sets.

Step 3: Now draw the matrix.

Branchase ⇒ | 1 | 2 | 3 | 4 | 5 | |

Cutsets |
||||||

C_{1} |
+1 | +1 | -1 | -1 | 0 | |

C_{5} |
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