Minimum Spanning Tree: The Cut Property | Baeldung on Computer Science

1. Overview

In this tutorial, we’ll discuss the cut property in a minimum spanning tree.

Furthermore, we’ll present several examples of cut and also discuss the correctness of cut property in a minimum spanning tree.

2. Definition of a Cut

In graph theory, a cut can be defined as a partition that divides a graph into two disjoint subsets.

Let’s define a cut formally. A cut $C = (S_1, S_2)$ in a connected graph $G(V, E)$ , partitions the vertex set $V$ into two disjoint subsets $S_1$ , and $S_2$ .

In graph theory, there are some terms related to a cut that will occur during this discussion: cut set, cut vertex, and cut edge. Before going further, let’s discuss these definitions here.

A cut set of a cut $C(S_1, S_2)$ of a connected graph $G(V, E)$ can be defined as the set of edges that have one endpoint in $S_1$ and the other in $S_2$ . For example the $C(S_1, S_2)$ of $G(V, E) = \{(i, j) \in E | j \in S_1, j \in S_2 \}$

A vertex $V_c$ is a cut vertex if there exists a connected graph $G(V, E)$ , and removing $V_c$ from $G$ disconnects that graph.

An edge $E_c$ is a cut edge of a connected graph $G(V, E)$ if $E_c \in E$ and $G - E_c$ disconnects the graph.

3. Example

In this section, we’ll see an example of a cut. We’ll also demonstrate how to find a cut set, cut vertex, and cut edge.

First, let’s take a look at a connected graph:

Here, we’ve taken $G(V, E)$ where $V=(V1, V2, V3, V4, V5, V6, V7)$ and $E=(E1, E2, E3, E4, E5, E6, E7, E8, E9)$ . Now let’s define a cut $C(S_1, S_2)$ in a $G$ :

So here, the cut $C$ disconnects the graph $G$ and divides it into two components $S_1$ and $S_2$ .

Now let’s discuss the cut vertex. According to the definition, the removal of the cut vertex will disconnect the graph. If we observe the graph $\mathbf{G}$ , we can see there are two cut vertices: $\mathbf{V3}$ and $\mathbf{V4}$ . Let’s verify this. First, we’re removing the vertex $V3$ from $G$ :

We can see the removal of vertex $V3$ disconnects the graph $G$ and breaks it into two graphs. Hence, we verified that $V3$ is a cut vertex in $G$ . Now let’s remove the vertex $V4$ from $G$ :

Again, when we remove $V4$ from $G$ , it disconnects the graph and creates two graphs. Hence, $V4$ is also a cut vertex in $G$ .

Let’s talk about the cut edge. According to the definition, if we remove a cut edge, it’ll disconnect the graph and results in two or more subgraphs. In $\mathbf{G}$ , it is easy to see that the edge $\mathbf{E4}$ is a cut edge. If we remove $E4$ from $G$ , it’ll break the graph $G$ into two subgraphs:

Next is the cut set. A cut set of a cut is defined as a set of edges whose two endpoints are in two graphs. Here, a cut set of the cut $\mathbf{C(S_1, S_2)}$ on $\mathbf{G}$ would be $\mathbf\{E4\}$ . We can see one endpoint of $E4$ belongs to $S_1$ and the other endpoint is in $S_2$ .

4. Variants of a Cut

There are two popular variants of a cut: maximum cut and minimum cut. In this section, we’ll discuss these two variants with an example.

Both minimum and maximum cut exist in a weighted connected graph. A minimum cut is the minimum sum of weights of the edges whose removal disconnects the graph. Similarly, a maximum cut is the maximum sum of weights of the edges whose removal disconnects the graph.

Let’s find the maximum and minimum cut:

Here we’re taking a connected weighted graph $G_1(V_1, E_1)$ . Also, we’ve defined 4 cuts in a graph $G_1$ .

So according to the definition, we’ll sum the weights of edges of each cut. Let’s start with $CUT 1$ . Total edge weight in $CUT 1 =$ sum of weights of $E1, E3, E10 = 2 + 2 + 4 = 8$ . In this way, the weight of $CUT 2, CUT 3,$ and $CUT 4$ would be $10, 6, 7$ .

Hence, the minimum cut is $\mathbf{CUT 3}$ with a weight of 6 and the maximum cut in $\mathbf{G_1}$ would be $\mathbf{CUT 2}$ as the sum of edge weight in $\mathbf{CUT 2}$ is greater than all other cuts in $\mathbf{G_1}$ .

5. Cut Property in Minimum Spanning Tree

5.1. Statement

Now we know that a cut splits the vertex set of a graph into two or more sets. A cut set contains a set of edges whose one endpoint is in one graph and the other endpoint is in another graph. When constructing a minimum spanning tree (MST), the original graph should be a weighted and connected graph. Let’s assume that all edges cost in the MST is distinct.

According to the cut property, if there is an edge in the cut set which has the smallest edge weight or cost among all other edges in the cut set, the edge should be included in the minimum spanning tree.

5.2. Example

We’re taking a weighted connected graph here:

In this example, a cut divided the graph $G$ into two subgraphs $S_1$ (green vertices) and $S_2$ (pink vertices). The cut set for $G$ would be $(E2, E3, E5, E6)$ . Now according to the cut property, the minimum weighted edge from the cut set should be present in the minimum spanning tree of $G$ . Here the minimum weighted edge from the cut set is $E5$ .

Now we’ll construct a minimum spanning tree of $G$ and check weather the edge $E5$ is present or not:

This is one of the minimum spanning trees of $G$ , and as we can see, the edge $E5$ is present here. So we can say the cut property works fine for the graph $G$ .

What about other graphs? Will the cut property holds for all other minimum spanning trees? Let’s find out in the next section.

5.3. Proof of Cut Property

Now to conclude that the cut property will work for all the minimum spanning tree, we’re presenting a formal proof in this section.

Let’s assume that we build a minimum spanning tree $T$ from a graph $G$ . We also defined a cut $C$ which split the vertex set into two sets $P$ and $Q$ . Furthermore, we assume that there exists an edge $E_C$ joining two sets $P$ , $Q$ , and has the smallest weight.

Now we’re starting this proof by assuming the edge $\mathbf{E_C}$ is not a part of the MST $\mathbf{T}$ . It would be interesting here to see what happens if we include $E_C$ to the MST $T$ . If we include $E_C$ in $T$ , it’ll create a cycle. But according to the definition of MST, a cycle can’t be part of MST.

Now, if we analyze the MST $T$ , there must be some edge in $T$ , let’s name it as $K$ , other than $E_C$ which has one endpoint in $P$ and another endpoint in $Q$ . Now initially, we assumed that $E_C$ has the smallest weight among all the edges which joins $P$ and $Q$ . This implies that the edge $\mathbf{K}$ must be of higher weight than $\mathbf{E_C}$ .

Therefore $T$ is a spanning tree but not a minimum spanning tree. If we include the edge $E_C$ and then construct the MST, the total weight of the MST $T$ would be less than the previous one. Also, $T$ can’t contain both $E_C$ and $K$ as it will create a cycle. Therefore our initial assumption that $E_C$ is not a part of the MST $T$ should be wrong.

Hence we can conclude that the minimum weighted edge in the cut set should be part of the minimum spanning tree of that graph.

Let’s simplify the proof with an example:

We’re taking a connected weighted graph $G$ . Now let’s define a cut $C$ of $G$ :

The cut $C$ divided the graph $G$ into two subgraphs $P$ and $Q$ . Now there are two edges that connect $P$ and $Q$ among which $E_C$ is the minimum weighted edge. First, we’ll construct a minimum spanning tree $T$ from $G$ without including the edge $E_C$ :

The total weight of the minimum spanning tree $T$ here is $19$ . Previously we defined that $E_C$ is the minimum weighted edge in the cut set. It means the weight of the edge $K$ should be greater than the edge $E_C$ . In such a case, the currently constructed spanning tree is not an MST as we can build a spanning tree which can be less weighted than the current one:

As we can see, when we include the edge $K$ in the spanning tree, the total weight of the spanning tree would become $19$ which is higher than the weight when we construct $T$ by including the edge $E_C$ . Therefore if we include the edge $\mathbf{K}$ , then it won’t be a minimum spanning tree.

Hence, we proved that the minimum spanning tree corresponds to a connected weighted graph should include the minimum weighted edge of the cut set.

5.4. Application

Shortest path algorithms like Prim’s algorithm and Kruskal’s algorithm use the cut property to construct a minimum spanning tree.

6. Conclusion

In this tutorial, we’ve discussed cut property in a minimum spanning tree.

We presented the correctness of the cut property and showed that cut property is valid for all minimum spanning trees.

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