Parameterized complexity of distances to sparse graph classes

This dissertation offers contributions to the area of parameterized complexity theory, which studies the complexity of computational problems in a multivariate framework. We demonstrate that for graph problems, many popular parameters can be understood as distances that measure how far a graph is from belonging to a class of sparse graphs. We introduce the term distance parameter for such parameters and demonstrate their value in several ways. The parameter tree-depth is uncovered as a distance parameter, and we establish several fixed-parameter tractability and hardness results for problems parameterized by tree-depth. We introduce a new distance parameter elimination distance to a class C. The parameter measures distance in a way that naturally generalises the notion of vertex deletion by allowing deletions from every component in each deletion step. We show that tree-depth is a special case of this new parameter, introduce several characterisations of elimination distance, and discuss applications. In particular we demonstrate that Graph Isomorphism is FPT parameterized by elimination distance to bounded degree, extending known results. We also show that checking whether a given graph has elimination distance k to a minor-closed class C is FPT, and, moreover, if we are given a finite set of minors characterising C, that there is an explicit FPT algorithm for this. We establish an algorithmic meta-theorem for several distance parameters, by showing that all graph problems that are both slicewise nowhere dense and slicewise first-order definable are FPT. Lastly, several fixed-parameter tractability results are established for two graph problems that have natural distance parameterizations.