Graphs: Cycles
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The document discusses various types of graph cycles, including Hamiltonian and Eulerian cycles and paths, defining them clearly and outlining relevant properties and theorems. It emphasizes the conditions under which Eulerian cycles and paths exist in connected graphs and presents Hierholzer's algorithm for finding Eulerian cycles. Additionally, it mentions the complexity of finding Hamiltonian cycles and references resources for further reading.
Graphs: Cycles
- 1. Graphs: Cycles Tecniche di Programmazione – A.A. 2012/2013
- 2. Summary A.A. 2012/2013Tecniche di programmazione2 Definitions Algorithms
- 4. Cycle A.A. 2012/2013Tecniche di programmazione4 A cycle of a graph, sometimes also called a circuit, is a subset of the edge set of that forms a path such that the first node of the path corresponds to the last.
- 5. Hamiltonian cycle A.A. 2012/2013Tecniche di programmazione5 A cycle that uses each graph vertex of a graph exactly once is called a Hamiltonian cycle.
- 6. Hamiltonian path A.A. 2012/2013Tecniche di programmazione6 A Hamiltonian path, also called a Hamilton path, is a path between two vertices of a graph that visits each vertex exactly once. N.B. does not need to return to the starting point
- 7. Eulerian Path and Cycle A.A. 2012/2013Tecniche di programmazione7 An Eulerian path, also called an Euler chain, Euler trail, Euler walk, or "Eulerian" version of any of these variants, is a walk on the graph edges of a graph which uses each graph edge in the original graph exactly once. An Eulerian cycle, also called an Eulerian circuit, Euler circuit, Eulerian tour, or Euler tour, is a trail which starts and ends at the same graph vertex.
- 8. Theorem A.A. 2012/2013Tecniche di programmazione8 A connected graph has an Eulerian cycle if and only if it all vertices have even degree. A connected graph has an Eulerian path if and only if it has at most two graph vertices of odd degree. …easy to check! Königsberg Bridges
- 9. Weighted vs. Unweighted A.A. 2012/2013Tecniche di programmazione9 Classical versions defined on Unweighted graphs Unweighted: Does such a cycle exist? If yes, find at least one Optionally, find all of them Weighted Does such a cycle exist? Often, the graph is complete If yes, find at least one If yes, find the best one (with minimum weight)
- 11. Eulerian cycles: Hierholzer's algorithm (1) A.A. 2012/2013Tecniche di programmazione11 Choose any starting vertex v, and follow a trail of edges from that vertex until returning to v. It is not possible to get stuck at any vertex other than v, because the even degree of all vertices ensures that, when the trail enters another vertex w there must be an unused edge leaving w. The tour formed in this way is a closed tour, but may not cover all the vertices and edges of the initial graph.
- 12. Eulerian cycles: Hierholzer's algorithm (2) A.A. 2012/2013Tecniche di programmazione12 As long as there exists a vertex v that belongs to the current tour but that has adjacent edges not part of the tour, start another trail from v, following unused edges until returning to v, and join the tour formed in this way to the previous tour.
- 13. A.A. 2012/2013Tecniche di programmazione13
- 14. A.A. 2012/2013Tecniche di programmazione14
- 15. A.A. 2012/2013Tecniche di programmazione15
- 16. Eulerian Circuits in JGraphT A.A. 2012/2013Tecniche di programmazione16
- 17. Hamiltonian Cycles A.A. 2012/2013Tecniche di programmazione17 There are theorems to identify whether a graph is Hamiltonian (i.e., whether it contains at least one Hamiltonian Cycle) Finding such a cycle has no known efficient solution, in the general case Example: the Traveling Salesman Problem (TSP)
- 18. A.A. 2012/2013Tecniche di programmazione18 Weighted or unweighted
- 19. What about JGraphT ? A.A. 2012/2013Tecniche di programmazione19 org.jgrapht.alg.HamiltonianCycle static <V,E> java.util.List<V> getApproximateOptimalForCompleteGraph (SimpleWeightedGraph<V,E> g) But… g must be a complete graph g must satisfy the “triangle inequality”: d(x,y)+d(y,z)<d(x,z)
- 20. A.A. 2012/2013Tecniche di programmazione20
- 21. Resources A.A. 2012/2013Tecniche di programmazione21 http://mathworld.wolfram.com/ http://en.wikipedia.org/wiki/Euler_cycle Mircea MARIN, GraphTheory and Combinatorics, Lectures 9 and 10, http://web.info.uvt.ro/~mmarin/
- 22. Licenza d’uso A.A. 2012/2013Tecniche di programmazione22 Queste diapositive sono distribuite con licenza Creative Commons “Attribuzione - Non commerciale - Condividi allo stesso modo (CC BY-NC-SA)” Sei libero: di riprodurre, distribuire, comunicare al pubblico, esporre in pubblico, rappresentare, eseguire e recitare quest'opera di modificare quest'opera Alle seguenti condizioni: Attribuzione — Devi attribuire la paternità dell'opera agli autori originali e in modo tale da non suggerire che essi avallino te o il modo in cui tu usi l'opera. Non commerciale — Non puoi usare quest'opera per fini commerciali. Condividi allo stesso modo — Se alteri o trasformi quest'opera, o se la usi per crearne un'altra, puoi distribuire l'opera risultante solo con una licenza identica o equivalente a questa. http://creativecommons.org/licenses/by-nc-sa/3.0/