Fast recognition of partial star products and quasi cartesian products

Abstract

This paper is concerned with the fast computation of a relation d on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis.A special case of d is the relation δ * , whose convex closure yields the product relation σ that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of d so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of δ * for graphs with maximum bounded degree.Furthermore, we define quasi Cartesian products as graphs with non-trivial δ * . We provide several examples, and show that quasi Cartesian products can be recognized in linear time for graphs with bounded maximum degree. Finally, we note that quasi products can be recognized in sublinear time with a parallelized algorithm. * We thank Lydia Ostermeier for her insightful comments on graph bundles, as well as for the suggestion of the term "quasi product". This work was supported in part by ARRS Slovenia and the Deutsche Forschungsgemeinschaft (DFG) Project STA850/11-1 within the EUROCORES Program EuroGIGA (project GReGAS) of the European Science Foundation. This paper is based on part of the dissertation of the third author.

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.…”

The Relaxed Square Property

“…Furthermore, if R is not well-behaved, this is equivalent to the existence of squares with two adjacent edges in same class ϕ ⊑ R and others in class(es) different from ϕ, see Figure 5 Let us now turn to the computational aspects of RSP-relations. It is an easy task to determine finest relations that have the square property in polynomial time, see [11,12]. In contrast, it seem to be hard in general to determine one or all finest RSP-relations.…”

Construction of k-matchings in graph products

“…Graphs and in particular graph products arise in a variety of different contexts, from computer science [3,16] to theoretical biology [32,37], computational engineering [11,12,13,14,17,18] or just as natural structures in discrete mathematics [10]. In this contribution, we are interested in the structure of maximum k-matchings in graph products that are based on k ′ -matchings in the factors.…”