Graph Theory - 2-Vertex-Connected Graphs

2-Vertex-Connected Graphs

A 2-vertex-connected graph (or biconnected) is a graph that remains connected even after the removal of any one of its vertices. In other words, there is no single vertex whose removal would disconnect the graph.

A graph G = (V,E) is 2-vertex connected if, for any two vertices u and v in G, there is a path between "u" and "v" that does not pass through any vertex other than "u" and "v". This means that the graph cannot be disconnected by removing a single vertex.

Properties of 2-Vertex-Connected Graphs

The 2-vertex-connected graph have several properties, such as −

• Presence of a Path: Between any two vertices in a 2-vertex-connected graph, there are at least two disjoint paths that do not share any vertices other than the endpoints. These are known as vertex-disjoint paths.
• No Cut Vertices: A 2-vertex-connected graph has no articulation points. Removing any single vertex will not disconnect the graph.
• Subgraph Connectivity: Any subgraph of a 2-vertex-connected graph that includes the original vertices (except for the removed vertex) will also be connected.
• Strong Connectivity: A 2-vertex-connected graph remains connected even if any one of its vertices is removed.

A cut vertex (or articulation point) in a graph is a vertex whose removal increases the number of connected components of the graph. In a 2-vertex-connected graph, no such cut vertex exists, which makes it a more strong structure for networks and other applications.

Conditions for 2-Vertex-Connectivity

A graph is 2-vertex-connected if and only if it satisfies the following conditions −

• It is connected, meaning there is a path between any pair of vertices in the graph.
• There are no cut vertices in the graph. Removing any vertex will not disconnect the graph.

For example, consider the graph below −

In this graph, removing any single vertex does not disconnect the graph. Therefore, the graph is 2-vertex-connected, as removing C and A vertices disconnects the graph.

Example: 2-Vertex-Connected Graph

The following graph is an example of a 2-vertex-connected graph −

In the above given graph, removing any single vertex does not disconnect the graph, but removing two vertices (e.g., C and E) will disconnect the graph.

Applications of 2-Vertex-Connected Graphs

2-vertex-connected graphs have various real-world applications, especially in fields where reliability and fault tolerance are important:

• Communication Networks: In communication networks, biconnected graphs ensure that there are multiple paths for data transmission. Even if a node fails, the network remains operational.
• Strong Infrastructure Design: In infrastructure systems such as transportation networks, power grids, and pipelines, biconnectivity guarantees that the failure of a single component does not lead to system failure.
• Distributed Computing: In distributed computing systems, 2-vertex-connected graphs ensure that the network of computers can still function even if one of the computers fails.
• Network Reliability: Biconnectivity is an important concept in designing networks that are strong to node failures and can continue to operate without significant loss of service.

Testing 2-Vertex-Connectivity

There are various ways to test if a graph is 2-vertex-connected, such as −

• DFS-Based Algorithm: By performing a DFS traversal and checking for articulation points, we can determine if the graph is biconnected.
• Minimum Degree Condition: A graph with a minimum degree of at least 2 and a connected structure is 2-vertex-connected. However, this condition is not always sufficient, as some graphs may still have articulation points.
• Edge-Connectivity Method: The edge-connectivity of a graph is the minimum number of edges that must be removed to disconnect the graph. If the edge-connectivity is greater than 1, the graph is likely 2-vertex-connected.

2-Vertex-Connectivity in Various Graphs

Following are the lists of various types of graph and its 2-vertex-connectivity −