Cut point, bridge and double connected components of undirected graphs

Concept

For undirected graph G, the number of connected components that are included in G after the deletion of the vertex V and its connected edge is said to be the joint point (articulation points) or cut point (cut points). Similarly, when you delete an edge E and its connected vertices, the graph contains more connected components, then E is the cut edge or bridge .

Cut-point formalization definition: A is a cut point when and only if there are two points u,v so that each path of U to V goes through a (after removing a, u to V does not have a path).

A graph without any cut points is called a double-connected graph . Any undirected graph is visualized as a combination of several maximal two-connected sub-graphs, each of which is called a two-connected component (bi-connected component).

The definition of two-connected components: is a large double-connected sub-graph, that is, if G is a double-connected component, then there is no g′, so that G is a g′ and g′ is also a dual-connected component.

point two connected components : A vertex is removed from the connected component, and the connected component is still connected. Special, the point double connected component is also called the block .

Side Two connected components : One edge is removed from the connected component, and the connected component is still connected.

Equivalence relation (equivalence relation): Edge E1 and