Cut vertex
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An undirected graph with n=5 vertices and n-2=3 cut vertices; the cut vertices (in red) are those not on either end

An undirected graph with no cut vertices

In mathematics and computer science, a cut vertex or articulation point is a vertex of a graph such that removal of the vertex causes an increase in the number of connected components. If the graph was connected before the removal of the vertex, it will be disconnected afterwards. Any connected graph with a cut vertex has a connectivity of 1.

While well-defined for directed graphs, cut vertices are primarily used in undirected graphs. In general, a connected, undirected graph with n vertices can have no more than n-2 cut vertices. Naturally, a graph may have no cut vertices at all.

A bridge is an edge analogous to a cut vertex; that is, the removal of a bridge increases the number of connected components of the graph.

Finding Cut vertices

A trivial O(nm) algorithm is as follows:

a = number of components in G (found using DFS/BFS)

for each i in V with incident edges

Remove i from V

b = number of components in G with i removed

if b > a

i is a cut vertex

restore i

An algorithm with the much better running time O(n+m) is known using depth-first search.

Cut vertices in trees

A vertex v of a tree G is a cut vertex of G if and only if the degree of the vertex is greater than 1.

See also

• Connectivity (graph theory)

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