1. CS120 Graphs

2. GRAPHS

3. Graph Types Undirected Graphs Weighted undirected graph

4. Graph Terminology
A graph and one of its subgraphs

5. Graph Terminology

6. Graph-like, but not Graphs

7. Running times for graph algorithms; expressed in terms of |V| and |E|.
If |E| is  |V2| then the graph is dense.
If |E| is  |V| then the graph is sparse.

8. Graphs as ADTs
What data structure would you use? What operations should be supplied?

9. Graphs as ADTs ADT graph operations Test if empty
Get number of vertices, edges in a graph
See if edge exists between two given vertices
Add vertex to graph whose vertices have distinct, different values from new vertex
Add/remove edge between two given vertices
Remove vertex, edges to other vertices
Retrieve vertex that contains given value

10. GRAPH REPRESENTATIONS

11. Adjacency Matrix Representation

12. Matrix Example 2

13. Matrix Example 3
A directed unweighted graph and (b) its adjacency matrix

14. Matrix Example4
A weighted undirected graph and (b) its adjacency matrix

15. Matrix representation requires too much storage for large graphs that are sparse, but very efficient for small graphs: use 1 bit per edge.

16. Adjacency List Representation

17. List Example 2

18. List Example 3
A directed graph and (b) its adjacency list

19. List Example 4
A weighted undirected graph and (b) its adjacency list

20. Adjacency List Representation

21. Vertex Degree/Indegree/Outdegree
What is the degree of vertex a? of vertex c?
What is the indegree of vertex 0 ? Vertex 2 ? What is the outdegree of vertex 3? Vertex 1?

22. Adjacency List Representation

23. Graph Traversals Visits all vertices it can reach
Visits all vertices if and only if the graph is connected
Connected component
Subset of vertices visited during a traversal that begins at a given vertex

24. Depth-First Search DFS traversal
Goes as far as possible from a vertex before backing up
Recursive traversal algorithm or stack based algorithm

25. Breadth-First Search BFS traversal
Visits all vertices adjacent to a vertex before going forward
BFS is a first visited, first explored strategy uses a queue based strategy
Contrast DFS as last visited, first explored

26. Depth-First Search
The results of a depth-first traversal, beginning at vertex a

27. Breadth-First Search
The results of a breadth-first traversal, beginning at vertex a

28. Applications of Graphs
Topological Sorting – finding an order of precedence relationships
Spanning Tree – find a connected component
Minimum Spanning Trees – find a minimum weight connected component
Shortest Paths – find the shortest path from source to destination, find all pairs shortest paths
Circuits – does a graph have a cycle
Some Difficult Problems – Traveling Salesman Problem, Hamiltonian Circuit Problem, Euler Circuit Problem, Finding a k-Clique in a graph, etc…..

29. Topological Sorting
A directed graph without cycles

30. Topological Sorting
The graph arranged according to two topological orders

31. Topological Sorting

32. Topological Sorting
Trace Depth First Search, output node to front of list when all its adjacent vertices have been visited.

33. Spanning Trees A tree is an undirected connected graph without cycles
Detecting a cycle in an undirected graph
Connected undirected graph with n vertices must have at least n – 1 edges
If it has exactly n – 1 edges, it cannot contain a cycle
With more than n – 1 edges, must contain at least one cycle

34. Spanning Trees
A spanning tree for the graph. Dotted edges not in spanning tree.

35. Spanning Trees
FIGURE Connected graphs that each have four vertices and three edges

36. Spanning Trees
Another possible spanning tree for the graph in

37. Spanning Trees
The BFS spanning tree rooted at vertex a for the graph

38. Minimum Spanning Trees
A weighted, connected, undirected graph