Computational aspects of the Helly property: a survey

Dourado, Mitre C.; Protti, Fábio; Szwarcfiter, Jayme Luiz

doi:10.1007/bf03192385

Cited by 12 publications

(2 citation statements)

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“…The class of pHelly hypergraphs was characterized by Berge and Duchet [1]. General (p, q, s)-Helly hypergraphs were characterized in [6]. Strong p-Helly hypergraphs were introduced by Golumbic and Jamison [7].…”

Section: Introductionmentioning

confidence: 99%

On the strong p-Helly property

Dourado¹,

Protti

Szwarcfiter

2008

Discrete Applied Mathematics

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The notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison in 1985 [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8-22]. Independently, other authors [A. Bretto, S. Ubéda, J. Žerovnik, A polynomial algorithm for the strong Helly property. , On maximal clique irreducible graphs. J. Combin. Math. Combin. Comput. 8 (1990) 187-193.] have also considered the strong Helly property in other contexts. In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p = 2, this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220]. We describe a characterization for this class and obtain an algorithm for recognizing such graphs. Again, the algorithm has polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p.

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Section: Introductionmentioning

confidence: 99%

On the strong p-Helly property

Dourado¹,

Protti

Szwarcfiter

2008

Discrete Applied Mathematics

Self Cite

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“…Let F be a family of subsets of a set S. We say that F satisfies the Helly property when every subfamily consisting of pairwise intersecting subsets has a non-empty intersection. The Helly property has an important role in Combinatorics and its computational aspects [5]. A graph is clique-Helly when its family of cliques satisfies the Helly property.…”

Section: Introductionmentioning

confidence: 99%