CST 504 Graphs
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The document discusses different concepts in graph theory including: - Euler paths and circuits which travel along each edge of a graph once. An Euler circuit is a closed path. - Hamiltonian paths and circuits which pass through each vertex of a graph once. - Weighted graphs where weights are assigned to edges to represent attributes like distance or cost. - Networks which are weighted graphs used to model real-world systems like transportation or computing. - Critical paths which identify the minimum time needed to complete interconnected tasks in a network.
CST 504 Graphs
- 2. The Königsberg Conundrum In the old city of Konigsberg there used to be only 5 bridges. People could take a round trip of all the bridges by crossing them only once. Go to page 70 in the text and trace with your finger the path you would take.
- 3. The Seven Bridges Then two more bridges were built. People tried but could not do a round trip and cross each bridge only once. Try this using the picture on the next slide.
- 4. Leonard Euler This smart guy, Leonard Euler, (pr. Oiler) was able to show why a round trip was impossible. He used dots to represent the land and lines to represent the bridges.
- 5. The Oiler does it! So Lenny showed that no matter where you started, you could not help but pass over a bridge two times. By doing this he introduced graph theory which shows how the elements of a set relate to each other.
- 6. First, Some Definitions graph Informally, a graph is a finite set of dots called vertices (or nodes) connected by links called edges (or arcs).
- 7. Definitions adjacent Two vertices are adjacent if they are connected by an edge.
- 8. Definitions Degree: The degree (or valence) of a vertex is the number of edge ends at that vertex. For example, in this graph all of the vertices have degree two.
- 9. Activities Book 2, p.18, Q. 1-7, some orally P. 19, Q. 9, 10, 14, 15
- 10. More Definitions complete graph A complete graph with n vertices is a graph with n vertices in which each vertex is connected to each of the others (with one edge between each pair of vertices). Here are the first five complete graphs:
- 11. Definitions connected A graph is connected if there is a path connecting every pair of vertices.
- 12. Activities P. 79, Q. 1-7 P. 81, Q. 8-11
- 13. Labyrinths So, the island in Königsberg and the bridges created a maze or a labyrinth. Labyrinths have existed for thousands of years. According to legend, King Minos created a labyrinth on the island of Crete. At the centre was his son, a half-man half- bull called a Minotaur (a/k/a Bob). If you could get out of the labyrinth before Bob got you, you survived.
- 14. Bob and His Maze
- 15. Bob Dies… Theseus kills Bob. He then escapes. He goes home.
- 16. Okay So that was not actually the maze. What did your graph look like?
- 17. Activities On page 85, do Q. 1-4 orally. On page 86, do Q. 6-10 On Page 87, do Q. 13, 16, 17, 21
- 18. Labyrinth Project You and a partner will pick a maze. Draw the labyrinth. Draw a graph of the labyrinth showing the vertices and the edges. This is a C1 task – math communication – constructs and uses networks of concepts; You will write a short paragraph about the labyrinth, how it works, and what you have learned.
- 19. Chartres Cathedral Prayer Maze
- 20. Brain Maze
- 21. Theseus Maze
- 22. Circular Maze
- 23. Euler Paths and Circuits Euler path: A path that travels over each edge once and only once in a connected graph. Euler Circuit: An Euler path that is closed.
- 24. Special Case An Euler path or circuit only exists if a graph has EXACTLY 2 vertices whose degrees are odd numbers. An Euler path exists when the degrees of all the vertices are even numbers.
- 25. Euler Circuit: Degree of Vertices
- 26. Drawing Euler Paths An Euler path must start at a vertex having an odd-numbered degree and end at another vertex with an odd- numbered degree. An Euler circuit can begin at any vertex and ends at the same vertex.
- 27. Hamiltonian Paths Hamiltonian Path: A path that passes through every vertex once and only once. Hamiltonian Circuit: A Hamiltonian path that finishes at the same vertex.
- 28. Distance of a Path D(P,Q) is the shortest distance between two points. Each edge/line is considered to have length one.
- 29. Activities Page 27, Q. 1-7
- 30. Tree Diagrams A tree diagram is a connected graph without a simple circuit. This can be used in planning jobs, electrical circuits and plumbing. A B C D
- 31. Directed Graph or Digraphs A directed graph is a graph in which each edge has a direction – called an arc; The arc has only one direction that can be followed; E.g one way street A path or circuit is SIMPLE if it contains no repeat arcs;
- 32. Weighted Graph A weighted graph is a graph, directed or not, in which a weight is attributed to each edge; The weight of a path is the sum of the weights of the edges that make up the path;
- 33. Activities Page 37, Q. 1, 3, 4 Page 41, Q. 13, 15, 16
- 34. Value of a Path The value of a path is the total of its weights. It can be a maximum or a minimum. To find the minimum, start with the path of lowest weight, and add additional edges until all the nodes are connected.
- 35. Networks A network is a graph in which every edge is assigned a weight. A weight can indicate time or cost. The weight of a path corresponds to the sum of the weights of the edges that make up the path. A network can be directed or undirected.
- 36. Chromatic Number The chromatic number is the minimum number of colours necessary to colour all of a graph’s vertices without any 2 adjacent vertices being of the same colour. It is also applied to maps.
- 37. Critical Path The critical path corresponds to a simple path of maximum value. Critical paths are used to determine the minimum amount of time required to carry out a task comprising several steps. To do this, you must know which steps are pre-requisites (needed ahead of time) for other steps, and which steps can be carried out at the same time.
- 38. Activity Page 52, Q. 1,2,5 Page 54, Q. 9, 10, 12, 13, 19