Edge connectivity

Definition

Edge connectivity is the minimum number of edges that need to be removed from a graph to make it disconnected or to separate it into different components. This concept is closely linked to how robust the network is, indicating how well it can withstand edge failures without losing connectivity. A graph with high edge connectivity remains intact even with the removal of several edges, while low edge connectivity suggests vulnerability.

5 Must Know Facts For Your Next Test

1. Edge connectivity can be defined for both directed and undirected graphs, but the calculations may differ.
2. If a graph is k-edge-connected, it means that at least k edges must be removed to disconnect the graph.
3. A complete graph has the maximum possible edge connectivity, equal to n-1, where n is the number of vertices.
4. Edge connectivity is particularly useful in network design, where robustness against failures is critical.
5. The value of edge connectivity provides insights into the reliability and redundancy of a network or graph structure.

Review Questions

• How does edge connectivity relate to network robustness and reliability?
  • Edge connectivity indicates how resilient a network is to edge failures. A network with high edge connectivity can sustain the loss of several edges without becoming disconnected, which is crucial for maintaining reliable communication in networks. Conversely, low edge connectivity signifies that even a small number of edge removals can significantly disrupt connections, highlighting vulnerabilities in the network structure.
• Compare and contrast edge connectivity with vertex connectivity in terms of their definitions and applications.
  • Edge connectivity refers to the minimum number of edges that must be removed to disconnect a graph, while vertex connectivity focuses on the minimum number of vertices needed for disconnection. Both concepts assess a graph's resilience but target different aspects—edge connectivity evaluates edge removal effects, while vertex connectivity looks at vertex impacts. In applications such as network design, understanding both metrics helps engineers optimize systems against potential failures.
• Evaluate the significance of edge connectivity in practical applications such as telecommunications or transportation networks.
  • Edge connectivity plays a critical role in designing robust telecommunications and transportation networks. High edge connectivity ensures that if some connections fail—due to natural disasters or maintenance—other routes remain available, allowing continued operation. This evaluation aids engineers in identifying weak points in the network and developing strategies to enhance reliability, thus ensuring uninterrupted service and minimizing downtime during disruptions.

Related terms

vertex connectivity: The minimum number of vertices that need to be removed to disconnect the remaining vertices in a graph.

cut-set: A set of edges whose removal increases the number of connected components in a graph.

connected graph: A graph in which there is a path between every pair of vertices, meaning it remains connected as long as no edges are removed.