Abstract
Abstract In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property. Say that a family of subsets has the Helly property when every subfamily of it, formed by pairwise intersecting subsets, contains a common element. There are many generalizations of this property which are relevant to some parts of mathematics and several applications in computer science. In this work, we survey computational aspects of the Helly property. The main focus is algorithmic. That is, we describe algorithms for solving different problems arising from the basic Helly property. We also discuss the complexity of these problems, some of them leading to NP-hardness results.
Highlights
In 1923, Eduard Helly [24, 57] published the famous theorem which originated the so called Helly property
The Helly property has been the object of studies in extremal hypergraph theory, as [87], and in other topics of the study of graphs
Since the Helly property and most variations considered in this work deal with the hyperedges of a hypergraph, isolated vertices are not relevant, and can be dropped
Summary
In 1923, Eduard Helly [24, 57] published the famous theorem which originated the so called Helly property. Since the Helly property and most variations considered in this work deal with the hyperedges of a hypergraph, isolated vertices are not relevant, and can be dropped. A hypergraph H is r-uniform when every hyperedge of H contains exactly r vertices. A hyperedge and a partial hypergraph of a graph G are respectively called edge and subgraph of G. Given a hypergraph H, the intersection graph, or line graph, of H is the graph containing one vertex for every hyperedge of H, and two vertices are adjacent if the corresponding hyperedges intersect.
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