Introduction to Hypergraphs
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Hypergraph is a generalization of a graph in which an edge can connect any number of vertices.
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Introduction to Hypergraphs
- 1. Introduction to Hypergraphs Ismael A. Ali iali1@kent.edu Semantic Web Meeting | KSU – Dept. of CS | Friday, April 24, 2015
- 2. Outline • Hypergraph Definition • Hypergraph Variations • Hypergraph Representation • Hypergraph Types • Undirected Hypergraphs • Directed Hypergraph • Transformations • Graph representation of Hypergraph • Hypergraph Properties • Applications of Hypergraph
- 4. • Hypergraph is a generalization of a graph in which an edge can connect any number of vertices. • Hypergraph H is a pair H = (V,E) where: • V is a set of elements called nodes or vertices, and • E is a set of non-empty subsets of V called hyperedges or edges. H=(X,E)
- 6. Empty, trivial, uniform, ordered and simple hypergraph • k-uniform hypergraph: When all hyperedges have the same cardinality; So a 2-uniform hypergraph is a classic graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. • A hypergraph is simple if all edges are distinct • An r-uniform hypergraph is said to be ordered if the occurrence of nodes in every edge is numbered from 1 to r.
- 8. Partial, and regular Hypergraph
- 9. Examples
- 12. Example#2
- 13. Example#2
- 14. Hypergraph Types 1. Undirected Hypergraphs 2. Directed Hypergraph
- 15. 1. Undirected Hypergraphs …. What mentioned before was undirected hypergraph
- 16. 2. Directed Hypergraphs (DH) DH has 2 models (of visualizing)
- 17. DH Model#1: Incidence digraph Dirhypergraph
- 18. DH Model#2:
- 21. • Transformations • Graph representation of Hypergraph 1. L(H) , Line Graph of Hypergraph 2. 2Sec(H) , 2 Section Graph of Hypergraph 3. Inc(H) , Incident Graph of Hypergraph
- 22. L(H) , Line Graph of Hypergraph Each Hyperedge in H is a vertex in L(H) The 2 vertices in L(H) are adjacent if their correspondent hyperedges in H has a common shared vertex
- 23. 2Sec(H) , 2 Section Graph of Hypergraph 2Sec(H) has the same vertices in H, The two vertices in 2Sec(H) are adjacent if the are in the same hyperedge of H
- 24. Inc(H) , Incident Graph of Hypergraph It is basically the bipartite graph of H, Where there is two disjoint sets V and E in each side of the Incident graph of H
- 25. Hypergraph Properties 1. Helly property 2. Conformality property
- 29. List of applications • Hypergraph Theory and System Modeling for Engineering • Chemical Hypergraph Theory • Hypergraph Theory for Telecommunications • Hypergraph Theory and Parallel Data Structures • Hypergraphs and Constraint Satisfaction Problems • Hypergraphs and Database Schemes • Hypergraphs and Image Processing • Distributed systems, databases, artificial intelligence • VLSI design • Directed hypergraphs can be very useful in many areas of sciences. Indeed directed hypergraphs are used as models in: • Formal languages. • Relational data bases. • Scheduling. The application papers are listed in the last chapter of this book
- 30. References Hypergraphs, Volume 45: Combinatorics of Finite Sets (North-Holland Mathematical Library) August 18, 1989 Hypergraph Theory: An Introduction (Mathematical Engineering) April 18, 2013 by Alain Bretto (Author)
- 31. Thank you.