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

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**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