Introduction to Hypergraphs
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This document introduces hypergraphs, which generalize graphs by allowing edges to connect any number of vertices. It defines hypergraphs and discusses variations such as k-uniform and simple hypergraphs. It also covers representations of undirected and directed hypergraphs, transformations between hypergraph and graph representations, properties like Helly and conformality, and applications in areas like engineering, chemistry, telecommunications, and more.
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.