# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## THE REPRESENTATION OF MULTI-HYPERGRAPHS BY SET INTERSECTIONS

Stanisław Bylka  and  Jan Komar

Institute of Computer Science
21 Ordona street, 01-237 Warsaw, Poland
e-mail: bylka@ipipan.waw.pl
e-mail: komjan@operamail.com

## Abstract

This paper deals with weighted set systems (V,E,q), where V is a set of indices, E ⊂ 2V and the weight q is a nonnegative integer function on E. The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,E,q) is defined to be an indexed family R = (Rv)v ∈ V of subsets of a set S such that
 ⎢⎢ ∩ v ∈ E Rv ⎢⎢ = q(E)   for eachE ∈ E.

A necessary condition for the existence of such representation is the monotonicity of q on E i.e., if F ⊂ E then q(F) ≥ q(E). Some sufficient conditions for weighted set systems representable by set intersections are given. Appropriate existence theorems are proved by construction of the solutions.

The notion of intersection multigraphs to intersection multi- hypergraphs - hypergraphs with multiple edges, is generalized. Some conditions for intersection multi-hypergraphs are formulated.

Keywords: intersection graph, intersection hypergraph.

2000 Mathematics Subject Classification: 05C62, 05C65.

## References

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