Algorithms and Complexity

In Theorem 1.3.1 we gave a characterization for Eulerian graphs: a graph G is Eulerian if and only if each vertex of G has even degree. This condition is easy to verify for any given graph. But how can we really find an Euler tour in an Eulerian graph? The proof of Theorem 1.3.1 not only guarantees that such a tour exists, but actually contains a hint how to construct such a tour. We want to convert this hint into a general method for constructing an Euler tour in any given Eulerian graph; in short, into an algorithm. In this book we generally look at problems from the algorithmic point of view: we want more than just theorems about existence or structure. As Lüneburg once said [Lue82], it is important in the end that we can compute the objects we are working with. However, we will not go as far as giving concrete programs, but describe our algorithms in a less formal way. Our main goal is to give an overview of the basic methods used in a very large area of mathematics; we can achieve this (without exceeding the limits of this book) only by omitting the details of programming techniques. Readers interested in concrete programs are referred to [SyDK83] and [NiWi78], where programs in PASCAL and FORTRAN, respectively, can be found.