Graphs and Paths: Drawing Task Table and Understanding Graph Theory Concepts

1. Task Tables Draw a graph for the following task table. Task Time Prerequisites Start 0 --- A 5 None B 1 None C 8 A D 2.5 A E 1.5 C F 3.5 C G 2 B, D H 1 E I 0.5 F, G, HFinish

2. Prerequisite – what needs to be done before a certain task Vertices – represent the tasks Edges – represent the person actually doing the task Arrows – show you what direction the tasks are being done in Number on the edge – goes with how much time it takes to do the task All graphs must have a Start and a Finish

3. A Start B

4. Critical Path – the list of tasks that were needed to show the shortest amount of time needed to complete a project Minimum project time – the shortest amount of time needed to complete the entire project

5. Earliest-start-time – the earliest EACH TASK can be started in the project Latest-start-time – the latest EACH TASK can be finished to stay on time for completion (procrastination date) These numbers are place at each task on your graph in a set of parentheses EST LST

6. Section 4.3 Vocabulary Connected – a graph where there is a path between each pair of vertices Adjacent – two vertices that are connected by an edge Complete – a graph where each pair of vertices is adjacent, denoted by Kn , where n is the number of vertices in the graph Degree – the number of edges that have a specific vertex as an endpoint Loop – an edge that begins and ends at the same vertex Multipleedge – more than one edge between two vertices

7. A C The graph is connected but not complete Points A and D would be an example of adjacent points B D E A B This graph is complete. (all complete graph must be connected) D C

8. Complete graphs K graphs 3 K graphs 2 We use K to denote a complete graph and the number tells us how many vertices.

9. Alternate Ways to Write Graphs Graph, Matrices, and List A B C D A 0 0 1 0 B 0 0 1 1 C 1 1 0 1 D 0 1 1 0 V = {A, B, C, D} E = {AC, BC, BD, CD} C A B D

10. C C B B D D A Degree: A = 1 B = 2 C = 3 D = 2 This is a multiple edge What is the degree of each vertex? This is called a loop

11. Each of the black boxes are doors. Is it possible to walk through every door exactly once? A D B E C F

12. Euler Circuits and Paths Making picture into a graph. The room are the vertices and the doors (or how you walk from room to room) are the edges. Since we can walk in either direction through a door the edges have no arrows on them.

13. Euler Circuits and Paths Euler Paths – to use each edge of the graph exactly once but to end a vertex different from the starting vertex. Euler Circuit – to use each edge of the graph exactly once and begin and end at the same vertex.

14. B • Euler Circuits, Paths or Neither. E A C C A B D G D A B F E E D C

15. Finding Circuits and paths To find a path – there must be exactly 2 vertices with odd degrees. They will be your beginning and end. To find a circuit – there must be a graph with all even vertices.

16. Digraphs Digraphs – graphs with directions, the arrows point in the only direction that you are able to travel on the edge (think one-way roads) Indegree – number of edge coming in to a vertex Outdegree – number of edges going out of a vertex

17. A Do these graphs have circuits, paths, or neither? B A B C E E G C D D F

18. Create a set of rules for finding a Euler circuit or path in a digraph. Path – Circuit -

19. Hamiltonian Circuits and Paths Is it possible to touch every vertex and return to the same vertex you started at? Is this possible on all graphs? F E D A C B

20. Hamiltonian Circuits and Paths Hamiltonian Paths – to use each vertex of the graph exactly once but to end a vertex different from the starting vertex. Hamiltonian Circuit – to use each vertex of the graph exactly once and begin and end at the same vertex. (You do not have to use all of the edges and you cannot repeat a vertex) • Cycle – any graph that is a Euler circuit and a Hamiltonian circuit

21. Do the following graphs have Hamiltonian Circuits, paths, or nethier? A D D E C B E A C B

22. Tournaments Tournament is a digraph that indicates who the winners are between two teams by the direction of the arrow. The arrow points at the losing team. In the above tournament A beats B, B beats no one, C beats A and B, and D beats everyone. B A D C

23. Tournament Vocabulary Transmitter – a vertex with a positive outdegree and a zero indegree Receiver – a vertex with a positive indegree and a zero outdegree.

24. Making a seating chart You are planning a trip and certain students do not get along with each other and cannot be in the same car. You also want to take the fewest number of cars possible. StudentsCannot sit with: Meghan Hannah, Brooke, Jenifer Ashley Hannah, Lisa, Brooke, Kelly Hannah Meghan, Ashley, Tiffany, Lisa Tiffany Hannah, Kelly Candace Brooke, Jenifer Lisa Ashley, Hannah Brooke Meghan, Ashley, Candace Jenifer Meghan, Candace Kelly Ashley, Tiffany

25. A M T K H J Ca B L

26. Graph Coloring This process does not mean that you must actually color a map. It is a way to organize and group item, such as the people and the number of cars needed for the trip in the previous example. The numbers of label (or colors) needed is referred to as the chromatic number.

27. E Copy the following map and color it so no two adjacent areas are the same color. A D F C B

28. E Make a graph from the following map and number the vertices, so that no two adjacent vertices have the same number. A D F C B