Thus, the biconnected components partition the edges of the graph; however, they may share vertices with each other.

Therefore, if the cycle double cover conjecture is true, every biconnected graph has a circular embedding.

The subgraphs formed by the edges in each equivalence class are the biconnected components of the given graph.

Therefore a biconnected graph has no articulation vertices.

In any biconnected graph with circuit rank , every open ear decomposition has exactly ears.

A biconnected component is a 2-connected component.

A biconnected component can be defined as a subgraph induced by a maximal set of nodes that has no separating vertex.

In the online version of the problem, vertices and edges are added (but not removed) dynamically, and a data structure must maintain the biconnected components.

The use of biconnected graphs is very important in the field of networking (see Network flow), because of this property of redundancy.

A planar graph is outerplanar if and only if each of its biconnected components is outerplanar.

More generally, the size of the longest cycle in an outerplanar graph is the same as the number of vertices in its largest biconnected component.

In the other direction, if the graph is not 2-vertex-connected, then it has an articulation vertex v separating some biconnected component of G from s and t.

Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph.

If the graph is not biconnected, its biconnected components may be colored separately and then the colorings combined.

In 2012, Taylor extended this algorithm to generate all permutations of cyclic edge-order for planar embeddings of biconnected components.

Every biconnected n-vertex cubic graph has O(2) 1-factorizations, and the prism graphs have Ω(2) 1-factorizations.

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

In graph theory, a biconnected component (also known as a block or 2-connected component) is a maximal biconnected subgraph.

The tree of the biconnected components Java implementation in the jBPT library (see BCTree class).

Robert Endre Tarjan and Jacobo Valdes (1980) used triconnected components for structural analysis of biconnected flow graphs.

Other contributions by Uzi Vishkin and various co-authors include parallel algorithms for list ranking, lowest common ancestor, spanning trees, and biconnected components.

The property of being 2-connected is equivalent to biconnectivity, with the caveat that the complete graph of two vertices is sometimes regarded as biconnected but not 2-connected.

For instance, the SPQR tree of a biconnected graph is a representation of the graph as a 2-clique-sum of its triconnected components.

The circular embedding conjecture or strong embedding conjecture states that every biconnected graph has a circular embedding onto a manifold.

Carsten Gutwenger and Petra Mutzel (2001) shared their practical experience on linear time computation of the triconnected components of biconnected graphs.