A Local Approach for Embedding Graphs into Cartesian Products

Abstract

The thesis is concerned with the graphs that have Cartesian product like structure, even if they are prime with respect to the product. In particular, we focus on heuristic methods that allow embedding of a given graph or at least its subgraph into a Cartesian product. For these purposes we use a local approach that covers a graph by small patches that reflect Cartesian product structure in the neighborhoods of single vertices.

The local approach provides a natural way to investigate approximate Cartesian products, because even if Cartesian product is disturbed, and thus prime w.r.t. the Cartesian product, we are able to recognize Cartesian structure in the parts of the graphs that were not disturbed. Hence, we design the algorithms that are able to recognize such parts of graphs and embed them into the Cartesian product. We thus improve the results of Hagauer and Žerovnik, who published an algorithm for the weak reconstruction of graphs. Our method promises to get answers also in other fields, such as Cartesian bundle recognition or recognition of Cartesian products of graphs with specified domain. Moreover, we show that our local approach can be very easily parallelized, which helps us to perform known algorithms in sublinear time.

Our research is motivated by practical applications in the fields of numerical methods, theoretical biology and engineering.