Bicomponents

Bicomponents are a concept primarily used in graph theory and related fields. A bicomponent of a graph is a maximal subgraph such that any two edges in the subgraph lie on a common simple cycle. Equivalently, a bicomponent is a maximal subgraph that remains connected after removing any single vertex. This second definition highlights their relationship to articulation points and bridges. If a graph has no articulation points or bridges, it is itself a single bicomponent.

The set of bicomponents partitions the edges of a graph. If a graph has no cycles, then each edge forms its own trivial bicomponent. When cycles are present, edges belonging to the same cycle will often be part of the same bicomponent. The study of bicomponents is closely related to the concept of biconnected components, which are maximal biconnected subgraphs. While the terms are often used interchangeably, there can be subtle differences in their precise definition depending on the context. Understanding bicomponents is crucial for analyzing graph connectivity and identifying critical nodes or edges within a network.