Hypergraph is a generalization of a graph in which an edge can connect any number of vertices.

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

3. Hypergraph Definition

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)

5. Hypergraph Variations

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.

7. Induced Sub-hypergraph

8. Partial, and regular Hypergraph

9. Examples

10. (Undirected) Hypergraph
Representation

11. (Undirected) Hypergraph Representation

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:

20. B-arcs and F-arcs

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

26. 1. Helly property

27. 2. Conformality property

28. Applications of Hypergraph

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.