K-connected graph

Definition

A k-connected graph is a type of graph where there are at least k vertex-disjoint paths between any two vertices. This means that if you remove up to k-1 vertices, the graph will still remain connected. This concept emphasizes the robustness and reliability of a graph, particularly in understanding paths and cycles as well as how connectivity can be affected by the removal of vertices.

5 Must Know Facts For Your Next Test

1. A graph is considered 1-connected if it remains connected as long as no vertices are removed, meaning it is connected in its original form.
2. In a k-connected graph, if you remove any k-1 vertices, there still exists at least one path connecting any pair of remaining vertices.
3. The concept of k-connected graphs is crucial in network design, where redundancy is necessary for reliability.
4. To determine if a graph is k-connected, one can utilize techniques such as finding all cut-sets or analyzing vertex cuts.
5. The maximum value of k for a given graph is known as its connectivity, which provides insight into how resilient the graph is to vertex removal.

Review Questions

• How does the definition of a k-connected graph relate to paths between vertices?
  • A k-connected graph ensures that there are at least k vertex-disjoint paths between any two vertices. This means that even if up to k-1 vertices are removed from the graph, at least one path will still exist between the remaining pairs of vertices. This property highlights how redundancy in paths contributes to overall connectivity and robustness within the graph.
• Discuss the role of cut vertices in determining the connectivity of a k-connected graph.
  • Cut vertices are critical points in a graph; their removal can lead to increased disconnected components. In a k-connected graph, there should be no cut vertex that reduces its connectivity below k when removed. Therefore, understanding cut vertices helps to identify vulnerabilities within graphs, showing how robust or fragile a given structure is when it comes to maintaining connectivity despite potential disruptions.
• Evaluate how the concept of k-connectivity can be applied to real-world networks such as telecommunications or transportation systems.
  • In real-world networks like telecommunications or transportation systems, applying k-connectivity helps ensure resilience against failures. For example, in a telecommunications network, having multiple pathways (k-disjoint paths) ensures that communication remains intact even if certain nodes (routers or switches) fail. By analyzing k-connectivity, engineers can design networks that withstand outages while maintaining operational efficiency, ultimately improving reliability and performance under adverse conditions.

Related terms

Vertex-disjoint paths: Paths between two vertices in a graph that do not share any common vertices, ensuring that the paths are completely separate.

Cut vertex: A vertex in a graph whose removal increases the number of connected components, indicating a critical point in the graph's structure.

Graph connectivity: A measure of how well the vertices in a graph are connected to each other, often assessed by the minimum number of vertices or edges needed to disconnect the graph.