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A. B. C. D. E. F. G. Topics. Paths and Circuits (11.2). e 2. e 1. e n. …. v 1. A. v. F. G. C. D. E. w. B. Definitions (p.667). Let G be a graph, and v and w be vertices of G. A walk from v to w has the form v e 1 v 1 e 2 v 2 ...e n-1 v n-1 e n w

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Topics

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Topics

  • Paths and Circuits (11.2)


Definitions p 667

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Definitions (p.667)

  • Let G be a graph, and v and w be vertices of G.

    • A walk from v to w has the form ve1v1e2v2...en-1vn-1enw

      where v0 (the starting point) is v and vn (the destination) is w.

      Note: Each vi and ei may be repeated.

Exercise: Find example walks from A to G in the graph.

Q: Is there a best walk? Shortest walk?


Why are we concerned with walks in a graph

Why are we concerned with walks in a graph?

  • Many real-world applications …

    • Navigation

    • Transportation

    • Computer networks

      Network topology

      Routing of data packets

      Wireless network (node movement)

  • Problem solving, games, gambling, …

  • Searching (e.g., searching the Internet)

  • Communication

  • Management


Why are we concerned with walks in a graph1

Why are we concerned with walks in a graph?

See http://en.wikipedia.org/wiki/Graph_theory#Applications

  • “Applications of graph theory are primarily, but not exclusively, concerned with labeled graphs and various specializations of these.

  • Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be represented by graphs.

  • The link structure of a website could be represented by a directed graph: the vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B.

  • A similar approach can be taken to problems in travel, biology, computer chip design, and many other fields.

  • The development of algorithms to handle graphs is therefore of major interest in computer science.”


Why are we concerned with walks in a graph2

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Why are we concerned with walks in a graph?

  • A walk may represent a solution in the problem domain.

  • Example: In a sociogram, a walk represents one of the communication paths between two persons in an organization or community.

  • With some specialization, concepts such as ‘channels of influence’, ‘most effective communication path’, ‘cliques’, etc. start to make sense.


Connectedness

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Connectedness

p.669: Let G be a graph.

  • Two vertices v and w of G are connected iff there exist a walk from v to w.

  • The graph G is connected iff given any two vertices v and w in G, there exist a walk from v to w.

    p.670:

    A graph H is a connected component of a graph G iff

  • H is a subgraph of G,

  • H is connected, and

  • No connected subgraph of G has H as a subgraph and contains vertices or edges that are not in H.

Example 11.2.4

Exercise: Find all the connected components in the example graph.


Paths

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Paths

  • Let G be a graph, and v and w be vertices of G.

    • A path from v to w is a walk from v to w with no repeated edges.

      Note:Repeated vertices are allowed.

Exercise: Find example paths from A to E in the graph.

Q1: How many paths are there?

Q2: What is the shortest path?


Simple paths

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Simple Paths

  • Let G be a graph, and v and w be vertices of G.

    • A simple path from v to w is a path from v to w with no repeated vertices.

      Note:Neither repeated edges nor repeated vertices are allowed.

Exercise: Find example simple paths from A to E in the graph.

Q1: How many simple paths are there?

Q2: What is the shortest simple path?


Closed walks

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Closed Walks

  • Let G be a graph, and v and w be vertices of G.

    • A closed walkis a walk that starts and ends at the same vertex.

      Note:Repeated edges and vertices are allowed.

Exercise: Find example closed walks in the graph.

Q1: How many closed walks are there? Does this question make sense at all?

Q2: Starting with node A, how many closed walks are there?


Circuits

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Circuits

  • Let G be a graph, and v and w be vertices of G.

    • A circuitis a closed walkwith no repeated edges.

      Note:Repeated vertices are allowed.

Exercise: Find example circuits in the graph.

Q1: Starting with node A, how many circuits are there?


Simple circuits

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Simple Circuits

  • Let G be a graph, and v and w be vertices of G.

    • A simple circuitis a circuitwith no repeated vertices.

      Note:Neither repeated edges nor repeated vertices are allowed.

Exercise: Find example simple circuits in the graph.

Q1: Starting with node A, how many simple circuits are there?


Comparisons

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Comparisons

  • p.667: Table of comparisons

  • Q: What’s the difference between a walk and a path?

  • How about a walk and a closed walk?

  • How about a path and a circuit?

  • How about a path and a simple path?

  • How about a circuit and a simple circuit?

  • How about a simple path and a simple circuit?


Questions

Questions?


Euler paths

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Euler Paths

  • p.675: Let G be a graph. An Euler path for G is a path that visits each edge exactly once.

    Note:Repeated vertices are allowed.

  • Exercise: Find an Euler path from A to D in the example graphs.

Q: Does every graph have an Euler path? Nope!

  • Theorem: In an Euler path, either all or all but two vertices (i.e., the two endpoints) have an even degree.


Finding an euler path

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Finding an Euler Path

  • Example 11.2.7

    Correction: Remove the edge between node I and K in the graph on page 676.

    Q: Find other Euler paths in the example graph.

    Q: How many Euler paths are there?

    Q: Is there an algorithm to find all the Euler paths in a given graph?

Exercise: Find all the Euler paths from A to D in this example graph.


Euler circuits

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Euler Circuits

  • p.671: Let G be a graph. An Euler circuit for G is a circuit that contains every vertex and every edge of G.

    Note: Although a vertex may be repeated, an edge may not be repeated in an Euler circuit.

    Exercise: Find an Euler circuit starting with A in the example graphs.

Theorem 11.2.2 (p.671): If a graph G has an Euler circuit, then every vertex of G has even degree.

Theorem 11.2.3 (p.672): If every vertex of a nonemptyconnected graph G has even degree, then G has an Euler circuit.


Questions1

Questions?


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