Graph Theory - 2-Edge-Connected Graphs

2-Edge-Connected Graphs

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

A graph G = (V,E) is 2-edge-connected if, for any two vertices u and v in G, there exists a path between u and v that does not pass through any edge whose removal would disconnect the graph.

Properties of 2-Edge-Connected Graphs

The 2-edge-connected graph has several properties, such as −

• Presence of Edge-Disjoint Paths: Between any two vertices in a 2-edge-connected graph, there are at least two disjoint paths that do not share any edges other than the endpoints. These are known as edge-disjoint paths.
• No Bridge Edges: A 2-edge-connected graph has no bridge edges (also called cut edges). Removing any single edge will not disconnect the graph.
• Subgraph Connectivity: Any subgraph of a 2-edge-connected graph that includes the original vertices (except for the removed edge) will also be connected.
• Strong Connectivity: A 2-edge-connected graph remains connected even if any one of its edges is removed.

A bridge edge (or cut edge) in a graph is an edge whose removal increases the number of connected components of the graph. In a 2-edge-connected graph, no such bridge edge exists, which makes it more strong in network design and other applications.

Conditions for 2-Edge-Connectivity

A graph is 2-edge-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 bridge edges in the graph. Removing any edge will not disconnect the graph.

For example, consider the graph below −

In this graph, removing any single edge does not disconnect the graph.

Example: 2-Edge-Connected Graph

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

In the above given graph, removing any single edge does not disconnect the graph, but removing two edges (e.g., C-D and E-F) will disconnect the graph.

Applications of 2-Edge-Connected Graphs

2-edge-connected graphs have various real-world applications, especially in fields where redundancy and fault tolerance are important −

• Communication Networks: In communication networks, 2-edge-connected graphs ensure multiple redundant paths for data transmission. Even if an edge fails, the network remains operational without significant disruptions.
• Network Design and Reliability: In transportation, power, and other infrastructure networks, 2-edge-connected graphs guarantee the system's strongness, preventing the failure of one component from affecting the entire network.
• Distributed Systems: In distributed computing, 2-edge-connected graphs ensure that the network remains functional even if a link or connection between nodes fails.
• Network Fault Tolerance: The concept of 2-edge-connectivity is important for designing fault-tolerant networks that can continue to operate even in the case of edge failures.

Testing 2-Edge-Connectivity

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

• DFS-Based Algorithm: By performing a DFS traversal and checking for bridge edges, we can determine if the graph is 2-edge-connected.
• Minimum Degree Condition: A graph with a minimum degree of at least 2 and a connected structure is likely to be 2-edge-connected. However, this condition is not always sufficient, as some graphs may still have bridge edges.
• 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-edge-connected.

2-Edge-Connectivity in Various Graphs

Following are the lists of various types of graphs and their 2-edge-connectivity −