Lecture 1-23-17

Articulation Points & Biconnectivity

• Articulation Point:
  • Node such that if the node is removed, the graph is no longer Connected

• G v. such that G

  v is disconnected

• Naive algorithm:
  • DFS/BFS on every node
  • Time: O(n(n + m))
• DFS
  • calculate as back up
• LOW

min(DFS#s of vertices adjacent to v, LOW(children of v))

if LOW(v) >= DFS#(parent(v))
    mark parent as articulation point

root: only AP if > 1 child

Biconnectivity

• Biconnected only if no articulation points in between

How to solve

• Sometimes better to solve subproblems on biconnected components

Connected

• Undirected
  • path from one vertex to every other vertex
  • path from all pairs of vertices
• Directed

Homework

• Connectivity for a digraph
• DFS on digraphs