Complexity of Graph Problems: Gonality, Colouring and Scheduling

In this thesis we study several graph problems. In the first part, we study the ‘gonality’ of graphs. This is a measure for the complexity of a graph. There are several variants of gonality. The divisorial gonality can be defined as a chip-firing game, the stable gonality is defined using morphims to trees. We show that it is NP-complete to compute the gonality of a graph. Moreover, we show that the stable gonality is computable. In the second part, we study a variant of graph colouring: rainbow vertex colouring. Here, we colour the vertices of a graph, such that every pair of vertices is connected by a rainbow path: a path in which all internal vertices have different colours. It is NP-hard to compute the minimum number of colours that is needed for such a colouring. We study this problem on several graph classes and give polynomial time algorithms. In the last part, we study a scheduling problem. In this problem several chains of jobs are given. Between every two consecutive jobs in a chain there is a specified amount of delay. Besides, every chain has a release date and deadline. The question is whether there exists a schedule that satisfies all constraints. We study several variants of this problem, and consider several parameters, such as the number of chains. All these variants are W[1]-hard, and some are even W[t]-hard for all t.