Connectivity of Graph

Whether it is possible to traverse a graph from one vertex to another is determined by how a graph is connected. Connectivity is a basic concept in graph theory that defines whether a graph is connected or disconnected. It has subtopics based on edge and vertex, known as edge connectivity and vertex connectivity.

Connectivity

A graph is said to be connected if there is a path between every pair of vertices. A graph with vertices that cannot reach each other is said to be disconnected.

Example 1: Connected Graph

In the following graph, it is possible to travel from one vertex to any other vertex. For example, one can traverse from vertex 'a' to vertex 'e' using the path a → b → e −

Example 2: Disconnected Graph

In the following graph, traversing from vertex 'a' to vertex 'f' is not possible because there is no path between them −

Edge Connectivity

The edge connectivity of a connected graph G is the minimum number of edges whose removal makes G disconnected. In other words, it is the number of edges in the smallest cut set of G.

Notation − λ(G)

If G has a cut edge (bridge), then λ(G) = 1.

Example

In the following graph, removing any single edge keeps it connected. But removing two specific edges disconnects it. Hence λ(G) = 2 −

Vertex Connectivity

The vertex connectivity of a connected graph G is the minimum number of vertices whose removal makes G either disconnected or reduces it to a trivial graph (single vertex).

Notation − K(G)

If G has a cut vertex, then K(G) = 1.

Relationship Between Connectivities

For any connected graph G, the following inequality always holds −

K(G) ≤ λ(G) ≤ δ(G)

Where K(G) is vertex connectivity, λ(G) is edge connectivity, and δ(G) is the minimum degree of any vertex in G.

Example

Calculate λ(G) and K(G) for a graph where δ(G) = 3 −

Given: δ(G) = 3

From the inequality: K(G) ≤ λ(G) ≤ δ(G) = 3

By inspection, K(G) ≥ 2 (no single vertex removal disconnects G)

Deleting edges {d,e} and {b,h} disconnects G:
    λ(G) = 2

Substituting: 2 ≤ λ(G) ≤ 3, and λ(G) = 2

From K(G) ≤ 2 and K(G) ≥ 2:
    K(G) = 2

Conclusion

Edge connectivity λ(G) is the minimum number of edges whose removal disconnects a graph, and vertex connectivity K(G) is the minimum number of vertices whose removal disconnects it. The relationship K(G) ≤ λ(G) ≤ δ(G) always holds for connected graphs.