1. Graph Search Applications

2. Application: Connected Components
Undirected graph – find connected components
Loop through all vertices, when you find one unvisited:
Run DFS or BFS to pull out all connected pieces
Mark the nodes as visited

3. Flood Fill Basic search over a grid, identifying connected components.
Explore current cell, then look at the 8 (or 4, or in 3D 8/26/etc.) adjacent ones recursively
Mark the cell by changing the value in the cell
Forms basis for lots of algorithms

4. Bipartite Graph Check Check whether bipartite (2-colorable)
Just run BFS (or DFS), coloring the nodes one of two colors as encountered.
If there is ever a conflict, then can’t be bipartite

5. Topological Sort
Given constraints about what has to come first (i.e. directed graph), find some ordering that is OK
Always start with a node that has in-degree of 0
Can modify DFS to do this
After visiting all adjacent nodes, add this node to the end of the list
Can modify BFS to do this
Add nodes with in-degree 0 to list
Pop a node off the list, remove all edges coming out
If removing an edge makes another node have in-degree 0, enqueue it.

6. Classifying Back Edges
Run DFS, but also keep a third state:
Unvisited (never started visiting)
Exploring (began processing neighbors, but not done)
Visited (visited and explored all neighbors)
A “Back Edge” goes from current node to an Exploring node (or a Visited node, if a directed graph)
Can be used to find cycles (back edges that aren’t a single undirected edge)

7. Articulation Points and Bridges
Find a vertex or an edge that when removed will disconnect the UNDIRECTED graph
Option 1: naïve approach
Calculate the number of connected components
Try, one-by-one, removing each edge/vertex
Recalculate number of CCs
If more CCs, then it’s an articulation point or bridge.

8. Articulation Points and Bridges
Find a vertex or an edge that when removed will disconnect the UNDIRECTED graph
Option 2: specialized algorithm
Compute DFS, but also store two values per node:
Num: the order in which the node was reached during DFS
Low: lowest Num reachable from current subtree
Update Low when you have a back edge
If a neighbor vertex has a Low value greater or equal to your own Num, then you’re at an articulation point
If Low(u) > Num (v), then u-v is a bridge
Likewise, if Low(v) > Num (u), then u-v is a bridge

9. Num
Low 1 2
Num is when visited
Low is lowest seen in
that subgraph 7 3 6 9 4 3 8 5 3

10. Num
Low 1 2
Articulation point if neighbor
has a low value >= your num 7 3 6 9 4 3 8 5 3

11. Num
Low 1 2
Bridge if Low(u) > Num(v)
for some edge u-v 7 3 6 9 4 3 8 5 3

12. Strongly Connected Components (Directed)
Notice: can replace a SCC by a single node, and result is a directed acyclic graph.
Perform DFS like before. SCCs are identified when a back edge creates a cycle (and therefore connects everything in that subtree.
Update Low only for Visited (not Exploring) vertices.
Add each node to a stack
When you finish a node where Num == Low, it marks the root of a SCC
Pop nodes off the stack until you get that node off; those nodes are a SCC