1. Graphs Examples on some basic graph concepts and definitions All graphics are taken from the LEDA demos: basic_graph_algorithms, gw_shortest_paths, graphwin
Geetika Tewari
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2. Vertex: The point of intersection of lines
Vertex: The point of intersection of lines. The terminating point of some lines in a figure or curve
Several Vertices
Edge: Line that connects 2 vertices
Loop: an edge that connects the same two vertices

3. Multiple Edges: More than one edge connecting two vertices

4. Undirected Graph: A graph whose edges are all undirected
Directed graph: A graph whose edges are directed

5. Simple Graph: A graph that does not contain any loops or multiple edges

6. Examples
1. Example of a graph: Notice it is undirected, has multiple edges and loops

7. 2. Example of a graph: Notice the unconnected components

8. 3. Example of a mutigraph (graph with multiple edges)

9. Special Cases of graphs
1. Complete Graph: A full graph where every edge is connected to every other edge
Complete graph of 3 vertices
Complete graph of 11 vertices

10. 2. Bipartite Graphs: A graph whose vertices can be divided into two classes. Edges exist only between vertices that belong to different classes.
An example of a complete bipartite graph

11. Paths
Path in an directed graph: a series of connected vertices
Notice in this example there is a path from vertex 0 to vertex 4: {0, 1, 3, 4}, and a path from vertex 0 to vertex 2:{0, 1, 3, 2}……

12. Path in an undirected graph
In this graph there is a path {11, 5, 7, 13, 10}
There are other paths too..

13. Hamilton Path in an Undirected Graph: A path that spans all the vertices in a graph
Hamilton Path in a Directed Graph: A path that spans all the vertices in the digraph
Eg: {0, 3, 5, 7, 10, 17, 21, 28}

14. Cycles: In a graph a path {Vo, V1, V2……Vk} forms a cycle if Vo = Vk and if V1, V2….Vk are distinct.
 There is a cycle in this undirected graph: {0, 1, 3, 0},
Cycles in this directed graph: {2, 3, 4, 2}

15. Hamilton Cycles: A Cycle that covers all the vertices in a graph
Hamilton Cycle in a directed graph {0, 1, 3, 2, 4, 5, 6, 7, 0}
Hamiltion Cycle in an undirected graph {7, 8, 1, 4, 3, 2, 0, 5, 6, 7}

16. Cyclic and Acyclic Digraphs
A cyclic digraph: a directed graph with cycles: {0, 3, 6, 0}, {0, 1, 2, 4, 5, 3, 6, 0}
An acyclic digraph: a directed graph with no cycles

17. Tree: A connected, acyclic undirected graph
Forest: An acyclic undirected graph, i.e. many trees grouped together

18. Articulation Points In an Undirected Graph
An articulation point is a vertex in a graph whose removal disconnects the graph. In the example below, the red vertices are the articulation points.