page_title: Biconnected component
section_title: Block-cut tree
hierarchical_section_title: Biconnected component / Related structures / Block-cut tree
context_page_description: In graph theory, a biconnected component is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph. The blocks are attached to each other at shared vertices called cut vertices or articulation points. Specifically, a cut vertex is any vertex whose removal increases the number of connected components.
context_section_description: A cutpoint, cut vertex, or articulation point of a graph G is a vertex that is shared by two or more blocks. The structure of the blocks and cutpoints of a connected graph can be described by a tree called the block-cut tree or BC-tree. This tree has a vertex for each block and for each articulation point of the given graph. There is an edge in the block-cut tree for each pair of a block and an articulation point that belongs to that block.
caption: A graph, and its block-cut tree.
  The blocks are: b1=[1,2], b2=[2,3,4], b3=[2,5,6,7], b4=[7,8,9,10,11], b5=[8,12,13,14,15], b6=[10,16] and b7=[10,17,18];
  cutpoints are: c1=2, c2=7, c3=8 and c4=10.