biconnected component

A biconnected component is a 2-connected component.

It is known from graph theory that the biconnected components and the separating vertices of a graph form a tree.

The width of a decomposition is the maximal number of vertices in a biconnected component.

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

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

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

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.

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

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.

A biconnected component of a graph is a maximal set of its nodes whose induced subgraph is connected and does not have any separating vertex.

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

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

Block graphs are sometimes erroneously called "Husimi trees", but that name more properly refers to cactus graphs, graphs in which every nontrivial biconnected component is a cycle.

The graphs of branchwidth 2 are the graphs in which each biconnected component is a series-parallel graph; the only minimal forbidden minor is the complete graph K on four vertices.

For other graphs, it may be optimized or reduced separately for each biconnected component of the graph before combining the solutions, as these components may be drawn so that they do not interact.

Indeed, a graph has treewidth at most 2 if and only if it has branchwidth at most 2, if and only if every biconnected component is a series-parallel graph.

The classic sequential algorithm for computing biconnected components in a connected undirected graph due to John Hopcroft and Robert Tarjan (1973) runs in linear time, and is based on depth-first search.

For instance, the series-parallel graphs are a subfamily of the partial 2-trees, and more strongly a graph is a partial 2-tree if and only if each of its biconnected components is series-parallel.

This property can be tested once the depth-first search returned from every child of v (i.e., just before v gets popped off the depth-first-search stack), and if true, v separates the graph into different biconnected components.

B(G) is necessarily a block graph: it has one biconnected component for each articulation vertex of G, and each biconnected component formed in this way must be a clique.

A graph has Hadwiger number at most three if and only if its treewidth is at most two, which is true if and only if each of its biconnected components is a series-parallel graph.

This can be represented by computing one biconnected component out of every such y (a component which contains y will contain the subtree of y, plus v), and then erasing the subtree of y from the tree.

The maximal planar graphs without separating triangles that may be formed by repeated splits of this type are sometimes called blocks, although that name has also been used for the biconnected components of a graph that is not itself biconnected.

Every constraint can be enforced on a node of the tree because each constraint creates a clique on its variables on the primal graph, and a clique is either a biconnected component or a subset of a biconnected component.