A local prime factor decomposition algorithm

Abstract

This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived.Moreover, since most graphs are prime although they can have a product-like structure, also known as approximate graph products, the practical application of the well-known "classical" prime factorization algorithm is strictly limited. This new PFD algorithm is based on a local approach that covers a graph by small factorizable subgraphs and then utilizes this information to derive the global factors. Therefore, we can take advantage of this approach and derive in addition a method for the recognition of approximate graph products. the classical PFD approach to this problem is strictly limited, since almost all graphs are prime, although they can have a product-like structure. In fact, even a very small perturbation, such as the deletion or insertion of a single edge, can destroy the product structure completely, modifying a product graph to a prime graph [4,45].The recognition of approximate products has been investigated by several authors, see e.g. [5,13,14,28,45,26,42,44,15,20,16,23]. In [28] and [45] the authors showed that Cartesian and strong product graphs can be uniquely reconstructed from each of its one-vertex-deleted subgraphs. Moreover, in [29] it is shown that k-vertex-deleted Cartesian product graphs can be uniquely reconstructed if they have at least k + 1 factors and each factor has more than k vertices. A polynomial-time algorithm for the reconstruction of one-vertex-deleted Cartesian product graphs is given in [7]. In [26,42,44] algorithms for the recognition of so-called graph bundles are provided. Graph bundles generalize the notion of graph products and can also be considered as approximate products.Another systematic investigation into approximate product graphs showed that a further practically viable approach can be based on local factorization algorithms, that cover a graph by factorizable small patches and attempt to stepwisely extend regions with product structures. This idea has been fruitful in particular for the strong product of graphs, where one benefits from the fact that the local product structure of neighborhoods is a refinement of the global factors [13,14]. In [13] the class of thin-neighborhood intersection coverable (NICE) graphs was introduced, and a quasi-linear time local factorization algorithm w.r.t. the strong product was devised. In [14] this approach was extended to a larger class of thin graphs which are whose local factorization is not finer than the global one, so-called locally unrefined graphs.In this contribution the results of [13] and [14] will be extended and generalized. The main result will be a new quasi-linear time local prime factorization algorithm w.r.t. the strong product that works for all graph classes. Moreover, this algorithm can be adapted for the recognition of approximate products. This new PFD algorithm is imp...

On the Cartesian skeleton and the factorization of the strong product of digraphs

“…Finally, since many graphs are prime although they can have a product-like structure, also known as approximate graph products, the aim is to design algorithms that can handle such "noisy" graphs. Most of the practically viable approaches are based on local factorization algorithms, that cover a graph by factorizable small patches and attempt to stepwisely extend regions with product structures [9,10,8,12,11]. Since the construction of the Cartesian skeleton works on a rather local level, i.e, the usage of neighborhoods, we suppose that our approach can in addition be used to establish local methods for finding approximate strong products of digraphs.…”

Fast recognition of partial star products and quasi cartesian products

“…Note, in [8] we considered "approximate products" which were first introduced in [7,6]. As approximate products are all graphs that have a (small) edit distance to a non-trivial product graph, it is clear that every bundle and quasi product can be considered as an approximate product, while the converse is not true.…”

Unique square property, equitable partitions, and product-like graphs

“…It can therefore be regarded as an "approximate graph product", albeit in a somewhat different sense than the deviations from product structures explored e.g. in [12,13,11].…”