BIOCONNECTED COMPONENT
• Articulation point: an articulation point in a connected graph is a vertex that, if delete, would break the graph into two or
more pieces (connected component).
• Biconnected graph: A graph with no articulation point called biconnected. In other words, a graph is biconnected if and
only if any vertex is deleted, the graph remains connected.
• Biconnected component: A biconnected component of a graph is a maximal biconnected subgraph- By maximal, we mean
that G contains no other subgraph that is both biconnected and properly contains . If a graph G is biconnected, then
G itself is called a block or a biconnected graph
• A graph that is not biconnected can divide into biconnected components, sets of nodes mutually accessible via two
distinct paths.
Articulation point - tarjan's algorithm
[Step 1.]
Find the depth-first spanning tree T for G
[Step 2.]
Add back edges in T
[Step 3.]
Determine DFN(i) and L(i)
DNF(i): the visiting sequence of vertices i by depth first search
L(i): the least DFN reachable from i through a path consisting of zero or more
tree edges followed by zero or one back edge
[Step 4.]
Therefore, we can say that u is an articulation point iff u is either the root of the
spanning tree and has two or more children, or u is not the root and u has a
child w such that low (w) >= dfn (u)
Step 1: Finding the depth-first spanning tree
Step 2: add back edges in T
Observations
• The root node of DFT is an articulation point if it has at least 2 children
• All leaf nodes are not articulation point
• 2,3,4,5,7 nodes can be
Step 3: Determine dfn(i) and low(i)
Step 4: low (w) >= dfn (u)
Here 2,3 5 are the articulation points