Biconnected Components
CS 312

Objectives
• Formulate problems as problems on graphs
• Implement iterative DFS
• Describe what a biconnected component is
• Be ready to start project 2

Problems on Graphs
• What to store in each node of the graph?
• What does it mean for two nodes to be connected in the graph?
• What property do I want to analyze about the graph?
– Then select or adapt an algorithm.

Generic graph exploration
Explore (G, v)
Input: G = (V,E) and v in V
Output: visited(u) is true u for all u reachable from v. visited(v) = true previsit(v) for each edge (v,u) E if not visited(u) then explore (u) postvisit (v)

Iterative DFS version
ExploreIterDFS (G, v)
Input: G = (V,E) and v in V
Output: visited(u) is true u for all u reachable from v. stack.push(v) if (stack.notEmpty()) v = stack.top
visited(v) = true previsit(v) for the next edge (v,u) E if not visited(u) then stack.push(u) if all edges from v have been traversed then stack.pop(v) postvisit (v)

Iterative DFS version
ExploreIterDFS (G, v)
Input: G = (V,E) and v in V
Output: visited(u) is true u for all u reachable from v. stack.push(v) if (stack.notEmpty()) v = stack.top
visited(v) = true previsit(v) for the next edge (v,u) E if not visited(u) then stack.push(u) if all edges from v have been traversed then stack.pop(v) postvisit (v)

Definitions
• Biconnected component
– The largest set of edges such that…
– There is a set of EDGES for which there is a simple cycle between every edge
– Or it is a single edge
• Bridge
– Bicconnected component of size 1
• Separating vertex
– Vertex which can be deleted to partition the graph.

a f e d g c b
Example

a f e d g c b
Example

DFS Tree

“Low” numbering
• “pre” ordering
• “low” numbering

Separating Vertices
How does that translate into a test on low and pre?

The Idea for the Algorithm
• If you can find low and pre, you can identify separating vertices.
– You can use DFS to find low and pre.
– Pre is easy.
• If you can find separating vertices, then you can use a second DFS to find biconnected components.
– Keep in mind that the biconnected components are edges not vertices.

Example f e d g h a c b a b g c d h f e ab bg gc cd dg  pop

f e d g h a c
Example low ( u )
 min pre ( u ) pre ( w ) where (v,w) is a backedge for a descendant v of u a,1 pre numbering.
b,2 b g,3 c,4 d,5 h,6 e,7 f,8

f e d g h a c
Example low ( u )
 min pre ( u ) pre ( w ) where (v,w) is a backedge for a descendant v of u a,1,1 low numbering.
b,2,1 b g,3,3 c,4,3 e,7,3 d,5,3 f,8,3 h,6,6 u is a sep. vertex iff there is a child v of u s.t.
pre(u) “< or =“ low(v)

Example low ( u )
 min pre ( u ) pre ( w ) where (v,w) is a backedge for a descendant v of u a a,1,1 f e d g c b c,4,3 b,2,1 g,3,3 e,7,3 h,6,6 d,5,5 c,4,4 g,3,3 b,2,2 a,1,1 d,5,5 f,8,8 h
Computing low.
Always initialize u.low = u.pre
h,6,6
(vertex, pre, lowest low so far)
When follow edge (u,v) and v is visited (and v != the DFS parent of u), u.low = min (u.low, v.pre)
When pop u off DFS stack, pass low to parent.
If u’s low is lower than it’s parent, then update parent.

Example f e d g a c low ( u )
 min pre ( u ) pre ( w ) where (v,w) is a backedge for a descendant v of u a,1,1 b,2,1 d is visited but, d is h’s dfs parent b c,4,3 g,3,3 d,5,5 h,6,6 d,5,5 c,4,4 g,3,3 b,2,2 a,1,1 d,5,5 e,7,3 f,8,8
Do nothing.
(vertex, pre, lowest low so far) h
Computing low.
Always initialize u.low = u.pre
h,6,6
When follow edge (u,v) and v is visited, u.low = min (u.low, v.pre)
When pop u off DFS stack, pass low to parent.
If u’s low is lower than it’s parent, then update parent.

Example low ( u )
 min pre ( u ) pre ( w ) where (v,w) is a backedge for a descendant v of u a f e d g c b c,4,3 a,1,1 b,2,1 g,3,3 e,7,3 d,5,5 h,6,6 d,5,5 c,4,4 g,3,3 b,2,2 a,1,1 d is visited but, d is h’s dfs parent
Do nothing.
d,5,5 f,8,8 h
Computing components
2 nd DFS h,6,6 use a stack of edges. push a mark onto the edge stack with the child of a SV when pop the child of an SV (in the node stack) pop back to the mark.
(vertex, pre, lowest low so far)