1. Topics
Paths and Circuits (11.2) A B C D E F G

2. 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. en e2 e1 v v1 … w A B C D E F G
Exercise: Find example walks from A to G in the graph.
Q: Is there a best walk? Shortest walk?

3. 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 …

4. Why are we concerned with walks in a graph?
See
“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.”

5. Why are we concerned with walks in a graph?B C D E F G
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.

6. Connectedness A graph H is a connected component of a graph G iff
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. A B C D E F G
Example
Exercise: Find all the connected components in the example graph.

7. 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. A C B D E
Exercise: Find example paths from A to E in the graph.
Q1: How many paths are there?
Q2: What is the shortest path?

8. 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. A C B D E
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?

9. Closed Walks Let G be a graph, and v and w be vertices of G.
A closed walk is 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?

10. Circuits Let G be a graph, and v and w be vertices of G.
A circuit is a closed walk with no repeated edges.
Note: Repeated vertices are allowed. A C B D E
Exercise: Find example circuits in the graph.
Q1: Starting with node A, how many circuits are there?

11. Simple Circuits Let G be a graph, and v and w be vertices of G.
A simple circuit is a circuit with no repeated vertices.
Note: Neither repeated edges nor repeated vertices are allowed. A B C D E F
Exercise: Find example simple circuits in the graph.
Q1: Starting with node A, how many simple circuits are there?

12. 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? A B C D E F
How about a circuit and a simple circuit?
How about a simple path and a simple circuit?

13. Questions?

14. Euler Paths A C B D E
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. A C B D E
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.

15. 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? A C B D E
Exercise: Find all the Euler paths from A to D in this example graph.

16. 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. A C B D E A C B D E
Theorem (p.671): If a graph G has an Euler circuit, then every vertex of G has even degree.
Theorem (p.672): If every vertex of a nonempty connected graph G has even degree, then G has an Euler circuit.

17. Questions?