1. Subgraphs, Connected Components, Spanning Trees
Depth-First Search
2/4/2019 8:01 PM
Subgraphs, Connected Components, Spanning Trees
Depth-First Search

2. Subgraphs A subgraph S of a graph G is a graph such that
The vertices of S are a subset of the vertices of G
The edges of S are a subset of the edges of G
A spanning subgraph of G is a subgraph that contains all the vertices of G
Subgraph
Spanning subgraph
Depth-First Search

3. Non connected graph with two connected components
Connectivity
A graph is connected if there is a path between every pair of vertices
A connected component of a graph G is a maximal connected subgraph of G
Connected graph
Non connected graph with two connected components
Depth-First Search

4. Trees and Forests A (free) tree is an undirected graph T such that
T is connected
T has no cycles
This definition of tree is different from the one of a rooted tree
A forest is an undirected graph without cycles
The connected components of a forest are trees
Tree
Forest
Depth-First Search

5. Spanning Trees and Forests
A spanning tree of a connected graph is
a spanning subgraph that is a tree
A spanning tree
is not unique unless the graph is a tree
Spanning trees have applications to the design of communication networks
A spanning forest of a graph
is a spanning subgraph that is a forest
Graph
Spanning tree
Depth-First Search

6. Cycles How to detect that a graph has a cycle? Hint: BFS (or DFS)
Depth-First Search

7. Connectivity Determines whether G is connected
How do we know G is not connected?
Hint: BFS or DFS
Depth-First Search

8. Connected Components Computes the connected components of G
How do we find all the connected components?
Depth-First Search

9. BFS for an entire graph Algorithm BFS(G, s)
Queue  new empty queue
enqueue(Queue, s)
while Queue.isEmpty()
v  dequeue(Queue)
if getLabel(v) != VISITED
visit v
setLabel(v, VISITED)
neighbors  getAdjacentVertices(G, v)
for each w  neighbors if getLabel(w) != VISITED // “children”
enqueue(Queue, w)
The algorithm uses a mechanism for setting and getting “labels” of vertices and edges
Algorithm BFS(G)
Input graph G
Output labeling of the edges and partition of the vertices of G
for each u  G.vertices()
setLabel(u, UNEXPLORED)
for each v  G.vertices()
if getLabel(v) = UNEXPLORED
BFS(G, v)
Breadth-First Search

10. DFS for an Entire Graph
The algorithm uses a mechanism for setting and getting “labels” of vertices and edges
Algorithm DFS(G, v)
visit v
setLabel(v, VISITED)
neighbors  getAdjacentVertices(G, v)
for each w  neighbors if getLabel(w) != VISITED // “children” DFS(G, w)
Algorithm DFS(G)
Input graph G
Output labeling of the edges of G as discovery edges and back edges
for each u  G.vertices()
setLabel(u, UNEXPLORED)
for each v  G.vertices()
if getLabel(v) = UNEXPLORED
DFS(G, v)
Depth-First Search

11. Spanning Tree Computes a spanning tree of G
How do we find a spanning tree?
Depth-First Search

12. BFS Example unexplored vertex visited vertex unexplored edgeC B A E D L0 L1 F A
unexplored vertex A
visited vertex
unexplored edge
discovery edge
cross edge L0 L0 A A L1 L1 B C D B C D E F E F
Breadth-First Search

13. BFS Example (cont.) L0 L1 L0 L1 L2 L0 L1 L2 L0 L1 L2 C B A E D F C B A
Breadth-First Search

14. BFS Example (cont.) L0 L1 L2 L0 L1 L2 L0 L1 L2 C B A E D F A B C D E F
Breadth-First Search

15. DFS Example unexplored vertex visited vertex unexplored edgeB A C E A A
visited vertex
unexplored edge
discovery edge
back edge D B A C E D B A C E
Depth-First Search

16. DFS Example (cont.) D B A C E D B A C E D B A C E D B A C E
Depth-First Search

17. Spanning Forest Computes a spanning forest of G
How do we find a spanning forest?
Depth-First Search

18. Path finding Given a start vertex s and a destination vertex z
Find a path from s to z
Depth-First Search

19. Path finding in a maze Vertex intersection, corner and dead end Edge
Corridor
Depth-First Search

20. BFS and path finding
Hint: use the spanning tree
Depth-First Search

21. BFS and path finding Use the spanning tree Start from destination
Go to its parent
Until source is reached
Depth-First Search

22. BFS and path finding Use the spanning tree
Start from destination
Go to its parent
Until source is reached
The found path is the shortest in terms of number of edges --- Why?
Depth-First Search

23. DFS and path finding Use the spanning tree
Start from destination
Go to its parent
Until source is reached
The spanning tree could be large
Use DFS but faster, ideas?
Depth-First Search

24. Path Finding find a path between two given vertices v and z
Algorithm pathDFS(G, v, z, S)
setLabel(v, VISITED)
push(S, v)
if v = z
return S.elements()
path = null
neighbors  getAdjacentVertices(G, v)
for each w  neighbors
if getLabel(w) != VISITED // “child”
path = pathDFS(G, w, z, S)
if path != null // has path via a child
return path
pop(S, v) //no path via a child or no children
return null
find a path between two given vertices v and z
call DFS(G, v) with v as the start vertex
use a stack S to keep track of the path
When destination vertex z is encountered,
return the path as the contents of the stack
Depth-First Search

25. DFS vs. BFS Applications DFS BFS DFS BFS
Spanning forest, connected components, paths, cycles 
Shortest paths
Biconnected components (still connected after removing one vertex, algorithm not discussed) C B A E D L0 L1 F L2 A B C D E F
DFS
BFS
Breadth-First Search

26. DFS vs. BFS (cont.) Back edge (v,w) Cross edge (v,w) DFS BFS
w is an ancestor of v in the tree of discovery edges
Cross edge (v,w)
w is in the same level as v or in the next level C B A E D L0 L1 F L2 A B C D E F
DFS
BFS
Breadth-First Search