1. Graph Terms By Susan Ott

2. Vertices Here are 7 vertices without any edges Each Vertex is labeled a different color and number

3. Edges Here are some vertices which are connected by edges For example an edge is connecting the two vertices labeled 0.

4. Multiple Edges and Loops The graph below has loops on vertices 0 and 1. A loop indicates that a vertex is connected to itself The graph below has multiple edges between the vertices labeled 0. A multiple edge indicates that there is more than one connection between 2 vertices.

5. Undirected Graphs An undirected graph is a set of vertices and edges such that if there is a connection if a has an edge to b then b has an edge to a.

6. Directed Graph In a directed graph, it is not always true that if a goes to b then b goes to a. In a directed graph, edges are sometimes called arcs which use arrows to indicate in which direction they are going

7. Simple Graph A simple graph is a graph that does not include any loops or multiple edges

8. Multigraphs Multigraphs are graphs that contain multiple edges

9. Complete Graphs A complete graph is a graph in which every vertex is connected to every other vertex. If a complete graph has n vertices it will have (n x (n-1))/2 edges

10. Bipartite Graphs A bipartite graph is a graph where there are 2 sets of vertices. The sets are such that the vertices in the same set will never have an edge between them.

11. Path in an undirected graph A path between A to B exists when there is a way to go from A to B by following edges without going through the same vertex twice. A path has two end vertices of degree 1, and every other vertex has degree 2

12. Path in directed graph A path in a directed graph from A to B is a way of going from A to B by following the directed edges and going through no vertex more than once. A directed path has one vertex with one incoming edge, one vertex with one out going edge, and the rest of the vertices have one of each

13. Hamilton Path in undirected graph A Hamilton path is a path that incorporates every vertex of the graph.

14. Hamilton path in directed graph A Hamilton path in a directed graph is a path in a directed graph that incorporates every vertex

15. Cycle in Undirected Graph A cycle exists in an Undirected Graph if there is a way to go from a vertex back to itself.

16. Cycle in Directed Graph A cycle in a directed graph exists if there is a way of going from a vertex and back by following the arrows.

17. Hamilton Cycle in Undirected Graph A Hamilton Cycle in an undirected graph is a cycle that incorporates every vertex once.

18. Hamilton Cycle in directed graph A Hamilton cycle in a directed graph is a directed cycle that incorporates every vertex once.

19. Cyclic digraph A Cyclic digraph is a directed graph that contains one or more cycles

20. Acyclic digraph An acyclic digraph is a directed graph that contains no cycles whatsoever

21. Tree A tree is a graph that is both acyclic and connected If a tree contains n vertices, it will have n- 1 edges.

22. Forest A Forest is a graph which has many disjoint trees for its components

23. A Subgraph that is connected but not a connected component The graph to the right has a subgraph labeled with ones. That is to say if you took away all the vertices with 0s and their adjoining edges, you would be left with the graph of the 1s. This subgraph is connected since it has a path from every vertex to every other vertex, but is not a connected component since the subgraph is not connected to the rest of the graph

24. Credits This Presentation was made possible through the use of the Leda Library’s Basic Graph Algorithm Demo.