Hypergraph Products

In this work, new definitions of hypergraph products are presented. The main focus is on the generalization of the commutative standard graph products: the Cartesian, the direct and the strong graph product. We will generalize these well-known graph products to products of hypergraphs and show several properties like associativity, commutativity and distributivity w.r.t. the disjoint union of hypergraphs. Moreover, we show that all defined products of simple (hyper)graphs result in a simple (hyper)graph. We will see, for what kind of product the projections into the factors are (at least weak) homomorphisms and for which products there are similar connections between the hypergraph products as there are for graphs. Last, we give a new and more constructive proof for the uniqueness of prime factorization w.r.t. the Cartesian product than in [Studia Sci. Math. Hungar. 2: 285–290 (1967)] and moreover, a product relation according to such a decomposition. That might help to find efficient algorithms for the decomposition of hypergraphs w.r.t. the Cartesian product

Local Prime Factor Decomposition of Approximate Strong Product Graphs: Local Prime Factor Decompositionof Approximate Strong Product Graphs

In practice, graphs often occur as perturbed product structures, so-called approximate graph products. The practical application of the well-known prime factorization algorithms is therefore limited, since most graphs are prime, although they can have a product-like structure. This work is concerned with the strong graph product. Since strong product graphs G contain subgraphs that are itself products of subgraphs of the underlying factors of G, we follow the idea to develop local approaches that cover a graph by factorizable patches and then use this information to derive the global factors. First, we investigate the local structure of strong product graphs and introduce the backbone B(G) of a graph G and the so-called S1-condition. Both concepts play a central role for determining the prime factors of a strong product graph in a unique way. Then, we discuss several graph classes, in detail, NICE, CHIC and locally unrefined graphs. For each class we construct local, quasi-linear time prime factorization algorithms. Combining these results, we then derive a new local prime factorization algorithm for all graphs. Finally, we discuss approximate graph products. We use the new local factorization algorithm to derive a method for the recognition of approximate graph products. Furthermore, we evaluate the performance of this algorithm on a sample of approximate graph products