1. Chapter 11 Graphs and Trees This handout: Terminology of Graphs Eulerian Cycles

2. Terminology of Graphs A graph (or network) consists of – a set of points – a set of lines connecting certain pairs of the points. The points are called nodes (or vertices). The lines are called arcs (or edges or links). Example:

3. Graphs in our daily lives Transportation Telephone Computer Electrical (power) Pipelines Molecular structures in biochemistry

4. Terminology of Graphs Each edge is associated with a set of two nodes, called its endpoints. Ex: a and b are the two endpoints of edge e An edge is said to connect its endpoints. Ex: Edge e connects nodes a and b. Two nodes that are connected by an edge are called adjacent. Ex: Nodes a and b are adjacent. a b c e f

5. Terminology of Graphs: Paths A path between two nodes is a sequence of distinct nodes and edges connecting these nodes. Example: Walks are paths that can repeat nodes and arcs. a b

6. A little history: the Bridges of Koenigsberg “Graph Theory” began in 1736 Leonhard Eüler –Visited Koenigsberg –People wondered whether it is possible to take a walk, end up where you started from, and cross each bridge in Koenigsberg exactly once

7. The Bridges of Koenigsberg A D C B 12 4 3 7 6 5 Is it possible to start in A, cross over each bridge exactly once, and end up back in A?

8. The Bridges of Koenigsberg A D C B 12 4 3 7 6 5 Translation into a graph problem: Land masses are “nodes”.

9. The Bridges of Koenigsberg 12 4 3 7 6 5 Translation into a graph problem : Bridges are “arcs.” A C D B

10. The Bridges of Koenigsberg 12 4 3 7 6 5 Is there a “walk” starting at A and ending at A and passing through each arc exactly once? Such a walk is called an eulerian cycle. A C D B

11. Adding two bridges creates such a walk A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A 12 4 3 7 6 5 A C D B 8 9 Here is the walk. Note: the number of arcs incident to B is twice the number of times that B appears on the walk.

12. Existence of Eulerian Cycle 12 4 3 7 6 5 A C D B 8 9 The degree of a node is the number of incident arcs 6 4 4 4 Theorem. An undirected graph has an eulerian cycle if and only if (1) every node degree is even and (2) the graph is connected (that is, there is a path from each node to each other node).