Cut-point

Cut-point

A cut-point is a concept in graph theory that refers to a vertex in a graph whose removal increases the number of connected components in the graph. In other words, if a graph contains a cut-point, removing that vertex will result in the graph being split into two or more separate parts.

Definition

Formally, a cut-point in a graph G is a vertex v such that the removal of v from G increases the number of connected components in G. In other words, if G has k connected components before removing v, it will have k+1 connected components after removing v.

Properties

Cut-points have several interesting properties:

1. Connectivity: A graph without any cut-points is called a biconnected graph, meaning that there are at least two vertex-disjoint paths between any pair of vertices. On the other hand, a graph with cut-points is called a non-biconnected graph.

2. Bridge: A cut-point can be thought of as a special case of a bridge, which is an edge whose removal increases the number of connected components in the graph. In fact, every cut-point is associated with at least one bridge.

3. Articulation points: Cut-points are also known as articulation points, especially in the context of undirected graphs. Articulation points play a crucial role in network analysis, as they represent vulnerable points in a network where the failure of a single vertex can lead to the disconnection of the network.

Examples

Let's consider a simple example to illustrate the concept of cut-points. Suppose we have a graph G with the following vertices and edges:

Example graph with cut-point

In this graph, vertex C is a cut-point. If we remove vertex C, the graph will be split into two separate components: {A, B} and {D, E}. Therefore, C is a cut-point in this graph.

Applications

The concept of cut-points has various applications in different fields:

1. Network analysis: Cut-points are used to identify critical points in a network where the failure of a single vertex can lead to the disconnection of the network. This information is crucial for designing robust and fault-tolerant networks.

2. Graph algorithms: Cut-points are used in various graph algorithms, such as graph traversal algorithms (e.g., depth-first search) and graph connectivity algorithms (e.g., Tarjan's algorithm). These algorithms rely on the identification of cut-points to efficiently explore and analyze the structure of a graph.

3. Social network analysis: Cut-points can be used to identify influential individuals in a social network. By identifying individuals who act as cut-points, we can understand their role in connecting different communities within the network.

See Also

• Graph theory
• Connected component
• Bridge (graph theory)
• Articulation point

References

1. West, D. B. (2001). Introduction to Graph Theory (2nd ed.). Prentice Hall. 2. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press.