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Chapter 10 Graphs and Trees

Chapter 10 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 ).

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Chapter 10 Graphs and Trees

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  1. Chapter 10Graphs 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. a c b e f 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.

  5. Terminology of Graphs: Paths • A path between two nodes is a sequence of distinct nodes and edges connecting these nodes. Example: • Walksare 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. A B C D The Bridges of Koenigsberg 1 2 3 4 6 5 7 Translation into a graph problem: Bridges are “arcs.”

  10. A B C D The Bridges of Koenigsberg 1 2 3 4 6 5 7 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. A B C D Adding two bridges creates such a walk 3 8 1 2 4 6 5 9 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. A 3 8 1 2 4 B 6 5 9 C 7 D 4 Existence of Eulerian Cycle 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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