Notes #2 Weighted and Directed Graphs
Weighted Graphs
1. What is the shortest distance
from point a to e?
2. Name the shortest Euler path.
How long is it?
3. Is there an Euler Circuit?
Directed Graphs
•The graphs we have looked at so far have been _______________________.
•____________________ give the ___________ a _______________ that
must be traveled from one vertex to another.
•Think about the difference between a regular 2-way street (undirected) and a
way street (directed).
Adding direction to a graph changes the number of solutions.
Our first possible solution from yesterday was:
D,E,F,A,D,C,A,B,D
Now that the graph is directed, is that still a solution?
Which solutions work now that the graph is directed?
1-
Tournament Graphs
•Directed graphs are often used in looking at _______________. Specifically,
round-robin tournaments where each person/team plays every other person/team
with no ties.
•In a tournament graph, the vertices correspond to the players. The edge between
each pair of players is oriented from the _____________ to the ____________.
Tournaments
1.Assume that the vertex-edge graph below is a representation of a Mathlete
round-robin tournament.
2.The winner of a tournament is defined by any vertex that comes __________ in
a _____________.
Remember:
1. The arrow always points from
the winner to the loser of the
game.
2. A Hamilton Path touches
every vertex once and only
once.
Examples
Which team wins the tournament?
Creating a Tournament Graph
There is a “HORSE” tournament between Shaq, Kobe, Dwayne, and Kevin. The
tournament is a round robin. Create a tournament graph that has Dwayne winning the
tournament and Shaq beating Kobe in their one-on-one game.
Step 1
Create a graph that has a vertex for every player.
Step 2
Mark the given information.
Step 3
The winner of the tourney is the start of a Hamilton
Path. So starting at Dwayne, make a directed path
that touches every vertex once. There are a few ways
you can do this.
Step 4
Fill in the other arrows carefully. Make sure the
“winner” has the most wins and that there aren’t any
other Hamilton paths that start with another player.