Bkomponenten

Bkomponenten, or B-components, is a term used in graph theory to denote maximal subgraphs that satisfy a robustness property, typically two-vertex connectivity. In many sources they are known as biconnected components or blocks. They partition the edges of a connected graph in a way that highlights critical structure, while articulation points (cut vertices) may belong to several B-components.

Definition and basic idea

A B-component of a connected graph is a maximal subgraph that is two-vertex-connected, meaning it remains connected after removing any single vertex. Equivalently, a B-component is a maximal subgraph that contains no articulation points within it. Each edge of the graph belongs to exactly one B-component, whereas articulation points can lie in multiple B-components, serving as connectors between them.

Properties and structure

The collection of B-components together with the cut vertices forms a block-cut structure, often represented as a block-cut tree. This representation illuminates how the global connectivity of the graph depends on a small number of critical vertices.

Computation and algorithms

B-components are typically found via a depth-first search (DFS) with lowlink values. A stack of edges is maintained; when the DFS backtracks to a point that satisfies a condition, the edges on the stack up to that point are popped to form a B-component. This standard approach runs in linear time with respect to the graph size, O(n+m).

Variants and related concepts

There are related notions such as edge-biconnected components (2-edge-connected components), which focus on edge removal rather than vertex removal. In some literature, B-components are synonymous with blocks or biconnected components.

Applications

B-components are used to analyze network reliability, identify critical nodes, simplify circuit designs, and study the resilience of complex systems by isolating robust subnetworks.