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Chapter 10 Graphs and Trees

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1 Chapter 10 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 Internet Social networks 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 1 2 3 B 4 C 5 6 7 D 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 1 2 3 B 4 C 5 6 7 D Translation into a graph problem: Land masses are “nodes”.

9 The Bridges of Koenigsberg
A 1 2 3 B 4 C 5 6 7 D Translation into a graph problem : Bridges are “arcs.”

10 The Bridges of Koenigsberg
A 1 2 3 B 4 C 5 6 7 D 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.

11 Adding two bridges creates such a walk
3 8 1 2 B 4 C 5 6 9 D 7 Here is the walk. A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A 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
4 1 2 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 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).


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