One answer is; a-g-c-b-f-e-i-k-h-d-j-a

a One answer is; a-g-c-b-f-e-i-k-h-d-j-a f e j g k i h Year 2 Warm Up State a Hamilton Circuit for the following figure: b d c

Directed and Weighted Graphs and Tournaments Objective: S2C4PO1

Weighted Paths and Circuits Weighted • _________ network paths and circuits have values along the edges. • To find the shortest network path you look for the path that travels the shortest distance but goes to all the vertices. • To find the shortest network circuit you look for the circuit that travels the shortest distance, remember a network circuit will start and end at the same point.

b • What is the shortest distance from point ato e? a 4 4 10 3 • Name the shortest Euler path. How long is it? d c 3 5 4 Ex: CBDCAGDEFG = 38 GDEFGACDBC = 38 All Euler Paths will have the same length. 6 e g 5 4 f 3. Is there an Euler Circuit? No Applications: Trucking Routes, Toll Roads, Fastest routes for running errands, Resistance on power lines

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 1-way street (directed). undirected graphs edges direction Directed graphs

B C D A E F Yesterday you found several different Euler Circuits in this figure. Let’s look at how adding direction to a graph changes the number of solutions we can find. 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? NO Which solutions work now that the graph is directed? ACDABDEFA ABDEFACDA

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 winner loser

Tournaments • Assume that the vertex-edge graph below is a representation of a Mathlete round-robin tournament. • The winner of a tournament is defined by any vertex that comes ____ in a _____________. first Hamilton Path Salim Amanda The arrow always points from the winner to the loser of the game. A Hamilton Path touches every vertex once and only once. Alessandro Amanda is the winner of the Tournament!!! Josie

D J K A D A 1 1 1 A X 0 1 0 X D 0 0 1 X J 0 0 0 X K K J Tournaments • You can also put the information in the graph into a matrix to find the winner. • A win is worth 1 and a loss is worth 0. = 3 = 1 = 1 = 0 A is the winner of the tournament!

ExamplesWhich team wins the tournament? Ex. 2 Ex. 1 Hamilton HS Team B Team C Team A Chandler HS Perry HS Perry HS GO PUMAS! Team E Team D Team D

Shaq Kobe Dwayne Kevin 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.

Shaq Kobe Dwayne Kevin Creating a Tournament Graph 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.

Summary… How do you determine the winner of a tournament using a vertex edge graph?

Homework: Directed Graph/Tournament WS #2

One answer is; a-g-c-b-f-e-i-k-h-d-j-a