Path decomposition of a graph has received an important amount of interest over the past decades because of its applications in algorithmic graph theory and in real life problems. For the computation of a path decomposition of small width, we use different heuritics approaches. One of the most useful method is by Bodlaender and Kloks. In this thesis, we focus on the computation, applications, transformation and approximation of a path decomposition of small width.
It is easy to convert a path decomposition in to nice path decomposition with same width, which is more convinent to use to find the graph parameters like independent sets, chromatic polynomials etc. Inspired by [28], we find an algorithm to compute the chromatic polynomial of a graph via nice path decomposition with small width.
Diese Arbeit beschäftigt sich mit dem Ising-Polynom, einem Graphenpolynom, das von einem physikalischen Modell abgeleitet ist. Es werden verschiedene Darstellungen des Polynoms, seine Beziehungen zu anderen Graphenpolynomen und in ihm enthaltene Grapheninvarianten vorgestellt. Weiter werden, insbesondere für spezielle Graphenklassen, Berechnungsmöglichkeiten beschrieben und der Rechenaufwand betrachtet.