Chapter 3 - Algorithmic graph theory

Abstract

This chapter introduces and relates applied combinatorial mathematics and algorithmic graph theory in the context of digital image processing. New definitions on the analogies with digital image processing are given. Graph theory is an important mathematical approach that can be used for mapping complex problems onto simple representations and models. It is based on a robust mathematical background that allows for the definition of optimal solution techniques for such problems. Thus, efficient algorithms can be derived, which can solve a particular problem based on its graph representation. Graph theory is an area of research by itself and finds applications in many fields, such as operational research and theoretical computer science. It also finds applications in image processing, where the discrete nature of image representations makes its use consistent. It is often the case that graph theory is implicitly used when developing a solution to a particular problem related to image processing. The aim here is to relate image processing concepts to algorithmic graph theory in order to take an advantage of this approach for further developments. The chapter presents terminology and definitions used in graph theory, followed by the summarization of well-known algorithms, which exist for the solution of problems defined in the context of graph theory. The classes of equivalent problems are defined and solutions are presented for the abstract representations of each class. A number of algorithms are presented for the shortest path problem and the minimum weighted spanning tree problem. These results are related to digital image processing concepts.