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Graphs_Euler_Circuit_Final_Presentation.pptx
- 1. Graphs and Euler'sCircuit Presentation with Sub-topic and Video | Due Next Week
- 2. Instructions Overview • 1.Topic: Graphs and Euler's Circuit • 2. Research beyond the blackboard • 3. Choose your own sub-topic related to the main topic • 4. Prepare a video related to the topic • 5. Present next week
- 3. Chosen Sub-topic: EulerPaths vs Euler Circuits • Euler Path: A path that uses every edge of a graph exactly once. • Euler Circuit: A circuit that uses every edge of a graph exactly once and starts and ends at the same vertex. • We'll compare these two and explore when each exists.
- 4. Examples and Explanation •Example of Euler Path: • - A graph with exactly two vertices of odd degree. • Example of Euler Circuit: • - A graph where all vertices have even degrees. • We'll use visual aids and simple graphs to show these. Sample Graph
- 5. Video Resource • Watchthis video to understand Euler Circuits and Paths in depth: • https://www.youtube.com/watch?v=GqX3r- DUdlY Sample Graph
- 6. Reminder • Make sureto: • - Understand the topic and sub-topic clearly • - Watch the video linked in the previous slide • - Prepare your part and be ready to present next week!
- 7. What is aGraph? • A graph is a set of vertices (nodes) connected by edges (lines). Graphs can represent networks like roads, social connections, or circuits.
- 8. Types of Graphs •1. Undirected Graph • 2. Directed Graph • 3. Weighted Graph • 4. Unweighted Graph
- 9. Degree of aVertex • The degree of a vertex is the number of edges connected to it. Vertices with even or odd degrees determine the existence of Euler paths and circuits. Sample Graph
- 10. Conditions for EulerPath • An Euler Path exists in a graph if there are exactly two vertices with odd degrees. Sample Graph
- 11. Conditions for EulerCircuit • An Euler Circuit exists if all vertices have even degrees. Sample Graph
- 12. Real-Life Example: Bridgesof Königsberg • Euler studied the problem of walking through all seven bridges of Königsberg without repeating a bridge. He proved it was impossible, founding graph theory.
- 13. Graph Representation • Graphscan be represented by: • 1. Adjacency Matrix • 2. Adjacency List
- 14. Euler Path Example1 • Graph with vertices A, B, C, D where A and D have odd degrees — has Euler Path.
- 15. Euler Circuit Example1 • Graph where all vertices have even degrees — has Euler Circuit starting and ending at the same vertex.
- 16. How to Identifyan Euler Path • 1. Count the degree of each vertex. • 2. If exactly two vertices have odd degrees, it's an Euler Path.
- 17. How to Identifyan Euler Circuit • 1. Count the degree of each vertex. • 2. If all vertices have even degrees, it's an Euler Circuit. Sample Graph
- 18. Eulerization • Eulerization isthe process of adding edges to a graph to make it Eulerian. Sample Graph
- 19. Hamiltonian vs Eulerian •Eulerian Path uses every edge once. • Hamiltonian Path visits every vertex once.
- 20. Directed Graphs andEuler Circuits • In directed graphs, Euler circuits exist if the in-degree equals out-degree for all vertices.
- 21. Why Learn EulerCircuits? • Euler circuits are used in planning, optimization, and solving routing problems like snow plow routes and garbage collection.
- 22. Quiz: Identify theType • Given a graph with vertex degrees: A(2), B(2), C(2), D(2) — Does it have an Euler Circuit or Path? • Answer: Euler Circuit.
- 23. Practice Problem • Drawa graph with 4 vertices that has an Euler Path but not an Euler Circuit.
- 24. Review and Recap •We've covered: Graph basics, Euler paths vs circuits, examples, conditions, real-life applications, and practice questions.