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- Paths and Circuits (11.2)

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- 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.

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

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

Q: Is there a best walk? Shortest walk?

- Many real-world applications …
- Navigation
- Transportation
- Computer networks
Network topology

Routing of data packets

Wireless network (node movement)

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- Problem solving, games, gambling, …
- Searching (e.g., searching the Internet)
- Communication
- Management
- …

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.”

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- 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.

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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.

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- 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.

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

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

Q1: How many paths are there?

Q2: What is the shortest path?

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- 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.

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

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?

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- 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.

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

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?

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- 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.

- A circuitis a closed walkwith no repeated edges.

Exercise: Find example circuits in the graph.

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

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- 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.

- A simple circuitis a circuitwith no repeated vertices.

Exercise: Find example simple circuits in the graph.

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

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- 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?

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- 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.

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- 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.

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- 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.