Chapter 9: Graphs

1. Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Chapter 9: Graphs Basic Concepts Lydia Sinapova, Simpson College

2. Graphs – Basic Concepts • Basic definitions: vertices and edges • More definitions: paths, simple paths, cycles, loops • Connected and disconnected graphs • Spanning trees • Complete graphs • Weighted graphs and networks • Graph representation • Adjacency matrix • Adjacency lists

3. Basic Graph Definitions A graph is a mathematical object that is used to model different situations – objects and processes: Linked list Tree (partial instance of graph) Flowchart of a program City map Electric circuits Course curriculum

4. Vertices and Edges Definition:A graph is a collection (nonempty set) of vertices and edges Vertices:can have names and properties Edges:connect two vertices, can be labeled, can be directed Adjacent vertices:there is an edge between them

5. A B C D C A B D Example Graph1 Vertices: A,B,C,D Edges: AB, AC, BC, CD Two ways to draw the same graph

6. A B C D A B C D Directed and undirected graphs Graph2 Graph3 These are two different graphs

7. A B C D More definitions : Path A list of vertices in which successive vertices are connected by edges A B C B A C D A B C A B C A B C D B A B A C

8. A B C D More definitions : Simple Path No vertex is repeated. A B C D D C A D C B A B A B C

9. A B C D More definitions : Cycle Simple path with distinct edges, except that the first vertex is equal to the last A B C A B A C B C B A C A graph without cycles is calledacyclic graph.

10. A B C D More definitions : Loop An edge that connects the vertex with itself

11. A B A B C C D D Connected and Disconnected graphs Connected graph: There is a path between each two vertices Disconnected graph : There are at least two vertices not connected by a path. Examples of disconnected graphs:

12. A B C D E Graphs and Trees Tree:an undirected graph with no cycles, and a node chosen to be the root Source graph:

13. C A B E D A C E B D Graphs and Trees Tree1: root A Tree2: root C

14. A B C D A spanning tree of an undirected graph A sub-graph that contains all the vertices, and no cycles.If we add any edge to the spanning tree, it forms a cycle, and the tree becomes a graph A B graph spanning tree C D

15. A B C D A A B B C C D D Examples All spanning trees of the graph on the previous slide

16. A B C D E Complete graphs Graphs with all edges present – each vertex is connected to all other vertices Dense graphs: relatively few of the possible edges are missing Sparse graphs: relatively few of the possible edges are present A complete graph

17. Weighted graphs and Networks Weighted graphs – weights are assigned to each edge (e.g. road map) Networks:directed weighted graphs (some theories allow networks to be undirected) B 2 1 C A 2 4 3 D

18. Graph Representation • Adjacency matrix • Adjacency lists

19. Adjacency matrix – undirected graphs Vertices: A,B,C,D Edges: AC, AB, AD, BD A B C D   A 0 1 1 1 B 1 0 0 1 C 1 0 0 1 D 1 1 0 0 The matrix is symmetrical A B C D

20. Adjacency matrix – directed graphs Vertices: A,B,C,D Edges: AC, AB, BD, DA A B C D  A 0 1 10 B 0 0 0 1 C 0 0 0 0 D 10 0 0 A B C D

21. Adjacency lists – undirected graphs Vertices: A,B,C,D Edges: AC, AB, AD, BD Heads lists A B C D B A D C A D A B A B C D

22. Adjacency lists – directed graphs Vertices: A,B,C,D Edges: AC, AB, BD, DA Heads lists A B C B D C = D A A B C D