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

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

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:

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

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

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

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

The Bridges of Koenigsberg A D C B Is it possible to start in A, cross over each bridge exactly once, and end up back in A?

The Bridges of Koenigsberg A D C B Translation into a graph problem: Land masses are “nodes”.

The Bridges of Koenigsberg Translation into a graph problem : Bridges are “arcs.” A C D B

The Bridges of Koenigsberg 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

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 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.

Existence of Eulerian Cycle A C D B 8 9 The degree of a node is the number of incident arcs 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).