Parameterized Graph Problems: Counting, the Tutte Polynomial and Logarithmic Space

In this thesis we study the computational complexity of various parameterized graph problems. The complexity of a problem measures the difficulty of a problem as the amount of time it costs to solve it. A parameterized problem includes some extra measure of the complexity of the instance, referred to as a parameter. Many of the problems we consider are counting problems, which can be thought of as asking for the number of solutions to some problem, rather than just a single solution. In the first part we study the Tutte polynomial, whose coefficients depend on some given graph. The Tutte polynomial generalizes a number of well known counting problems. We study a number of these in more detail, including graph colorings, connected edgesets and spanning forests. We give a classification of the parameterized complexity of computing the Tutte polynomial on integer points, for the parameters pathwidth, treewidth and cutwidth. In the second part we discuss the complexity classes XNLP and XALP, as well as introducing the counting variants #XLP and #XALP of these classes. A complexity class can be thought of as a large collection of algorithmic problems, that share some upper bound on their computational complexity. We give complete problems for all four of these classes, which is to say we show for some problems in the classes that they are at least as hard as any other problem in that class.