In the mathematical discipline of graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, a biconnected graph is a connected graph

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

 with no articulation vertices.

In other words, a biconnected graph is connected and nonseparable, meaning that if any vertex

Vertex (graph theory)

In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...

 were to be removed, the graph will remain connected.

The property of being 2-connected

K-vertex-connected graph

In graph theory, a graph G with vertex set V is said to be k-vertex-connected if the graph remains connected when you delete fewer than k vertices from the graph...

 is equivalent to biconnectivity, with the caveat that the complete graph

Complete graph

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...

 of two vertices is sometimes regarded as biconnected but not 2-connected.

This property is especially useful in maintaining a graph with a two-fold redundancy

Redundancy (engineering)

In engineering, redundancy is the duplication of critical components or functions of a system with the intention of increasing reliability of the system, usually in the case of a backup or fail-safe....

, to prevent disconnection upon the removal of a single edge (or connection).

The use of biconnected graphs is very important in the field of networking (see Network flow

Network flow

In graph theory, a flow network is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in Operations Research, a directed graph is called a network, the vertices are called nodes and the edges are...

), because of this property of redundancy.

Definition

A biconnected undirected graph is a connected graph that is not broken into disconnected pieces by deleting any single vertex (and its incident edges).

A biconnected directed graph

Directed graph

A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...

is one such that for any two vertices v and w there are two directed paths from v to w which have no vertices in common other than v and w.

Nonseparable (or 2-connected) graphs (or blocks) with n nodes

Vertices	Number of Possibilities
1	0
2	1
3	1
4	3
5	10
6	56
7	468
8	7123
9	194066
10	9743542
11	900969091
12	153620333545
13	48432939150704
14	28361824488394169
15	30995890806033380784
16	63501635429109597504951
17	244852079292073376010411280
18	1783160594069429925952824734641
19	24603887051350945867492816663958981