4,067 research outputs found
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
Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms
It is well-known that all finite connected graphs have a unique prime factor
decomposition (PFD) with respect to the strong graph product which can be
computed in polynomial time. Essential for the PFD computation is the
construction of the so-called Cartesian skeleton of the graphs under
investigation.
In this contribution, we show that every connected thin hypergraph H has a
unique prime factorization with respect to the normal and strong (hypergraph)
product. Both products coincide with the usual strong graph product whenever H
is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as
a natural generalization of the Cartesian skeleton of graphs and prove that it
is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian
skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can
be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree
and bounded rank
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