Cartesian product of hypergraphs: properties and algorithms

Erratrum: In the accepted version of this survey [36] it is mistakenly stated that the direct products ⌢ × and × and the strong product ⌢ ⊠ are associative. In [32], we gave counterexamples for these cases and proved associativity of the hypergraph productsAbstract. A surprising diversity of different products of hypergraphs have been discussed in the literature. Most of the hypergraph products can be viewed as generalizations of one of the four standard graph products. The most widely studied variant, the so-called square product, does not have this property, however. Here we survey the literature on hypergraph products with an emphasis on comparing the alternative generalizations of graph products and the relationships among them. In this context the so-called 2-sections and L2-sections are considered. These constructions are closely linked to related colored graph structures that seem to be a useful tool for the prime factor decompositions w.r.t. specific hypergraph products. We summarize the current knowledge on the propagation of hypergraph invariants under the different hypergraph multiplications. While the overwhelming majority of the material concerns finite (undirected) hypergraphs, the survey also covers a summary of the few results on products of infinite and directed hypergraphs. Mathematics Subject Classification (2000). Primary 99Z99; Secondary 00A00.

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“…for all x, y ∈ V 1 and e ⊆ V 1 . Every hypergraph is uniquely (up to isomorphism) determined by its L2-section and vice versa [11,10]…”

Section: L2-sectionsmentioning

confidence: 99%

“…The Cartesian product of hypergraphs has been investigated by several authors since the 1960s [37,38,11,9,10,17,49,53]. It is probably the best-studied construction.…”

Section: The Cartesian Productmentioning

confidence: 99%

A Survey on Hypergraph Products

2012

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“…If t(e) = h(e) for all e ∈ E the hypergraph is called undirected and directed, otherwise. Products of hypergraphs have been investigated by several authors since the 1960s [2,3,5,6,7,8,10,16,19,20,22,23,25,28,32]. It was shown by Imrich [19] that connected undirected hypergraphs have a unique prime factor decomposition (PFD) w.r.t.…”

Section: Introductionmentioning

confidence: 99%

Fast factorization of Cartesian products of (directed) hypergraphs

Hellmuth

Lehner

2016

Theoretical Computer Science

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Cartesian products of graphs and hypergraphs have been studied since the 1960s. For (un)directed hypergraphs, unique prime factor decomposition (PFD) results with respect to the Cartesian product are known. However, there is still a lack of algorithms, that compute the PFD of directed hypergraphs with respect to the Cartesian product.In this contribution, we focus on the algorithmic aspects for determining the Cartesian prime factors of a finite, connected, directed hypergraph and present a first polynomial time algorithm to compute its PFD. In particular, the algorithm has time complexity O(|E||V|r 2 ) for hypergraphs H = (V, E), where the rank r is the maximum number of vertices contained in an hyperedge of H. If r is bounded, then this algorithm performs even in O(|E| log 2 (|V|)) time. Thus, our method additionally improves also the time complexity of PFD-algorithms designed for undirected hypergraphs that have time complexity O(|E||V|r 6 ∆ 6 ), where ∆ is the maximum number of hyperedges a vertex is contained in.

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“…Products of hypergraphs have been well-investigated since the 1960s, see e.g. [1,2,3,4,6,7,9,10,12,13,14,15,18].…”

Section: Introductionmentioning

confidence: 99%

Associativity and Non-Associativity of Some Hypergraph Products

et al. 2016

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Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal rank preserving direct product, and the normal product. Counter-examples are given for the strong product as well as the non-rankpreserving and the maximal rank preserving direct product.

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Cartesian product of hypergraphs: properties and algorithms

Cited by 8 publications

References 12 publications

A Survey on Hypergraph Products

A Survey on Hypergraph Products

Fast factorization of Cartesian products of (directed) hypergraphs

Associativity and Non-Associativity of Some Hypergraph Products

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