Cut Set and Cut Vertex 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.

Connectivity

A graph is said to be connected if there is a path between every pair of vertices. A graph with multiple disconnected vertices and edges is said to be disconnected.

Cut Vertex

Let G be a connected graph. A vertex V ∈ G is called a cut vertex (or articulation point) of G if removing V (and all its incident edges) results in a disconnected graph. A connected graph G may have at most (n − 2) cut vertices.

Example

In the following graph, vertices 'e' and 'c' are the cut vertices −

By removing vertex 'e', there is no path between the left group {a, b, c, d} and the right group {f, g, h, i}, making the graph disconnected. Hence 'e' is a cut vertex. Similarly, removing 'c' would also disconnect the graph.

Cut Edge (Bridge)

Let G be a connected graph. An edge e ∈ G is called a cut edge (or bridge) if removing it results in a disconnected graph.

Example

In the same graph structure above, the edge (c, e) is a cut edge. Removing it breaks the graph into two disconnected components −

Note − For a connected graph G with n vertices −

• A cut edge e ∈ G exists if and only if 'e' is not part of any cycle in G.
• The maximum number of cut edges possible is n − 1.
• Whenever cut edges exist, cut vertices also exist (at least one vertex of a cut edge is a cut vertex).
• If a cut vertex exists, a cut edge may or may not exist.

Cut Set of a Graph

Let G = (V, E) be a connected graph. A subset E' of E is called a cut set of G if deleting all edges in E' from G makes G disconnected. A cut set must be minimal − no proper subset of E' should also disconnect G.

Example

For a graph with labeled edges e1 through e9, some possible cut sets are −

E1 = {e1, e3, e5, e8}     ? disconnects G
E3 = {e9}                  ? smallest cut set (single bridge edge)
E4 = {e3, e4, e5}          ? disconnects G

The smallest cut set E3 = {e9} contains just one edge, which means e9 is a bridge (cut edge).

Conclusion

A cut vertex disconnects a graph when removed, a cut edge (bridge) disconnects a graph when deleted, and a cut set is a minimal set of edges whose removal disconnects the graph. Cut edges are never part of any cycle, and the existence of cut edges guarantees the existence of cut vertices.