Parameterized complexity of computing twin width of a graph

Abstract

This thesis explores the design of efficient algorithms for determining the twin-width of a graph from a parameterized complexity perspective. Twin-width is a graph parameter that gives a decomposition of the graph in the lens of a contraction sequence. It is a parameter that measures the similarity of a graph to a cluster graph with smaller values of twin-width corresponding to more regular and structured graphs. However, determining the exact value of twin-width for a given graph is NP-complete. The thesis focuses on fixed-parameter tractable (FPT) algorithms, which can solve instances of the problem efficiently if the parameter remains small, or identifying suitable parameters that lead to meaningful hierarchy results. The thesis presents a detailed analysis of computing twin-width under various structural graph parameters and their relationship to twin-width, including twin cover number, vertex cover number, neighborhood diversity, edge deletion distance to cluster graph, and restricted modular partitions. The thesis makes several contributions to the field, including the development of fixed-parameter tractable (FPT) algorithms for computing twin-width parameterized by these graph parameters. These algorithms are complemented with reduction rules that simplify the input graph, making the algorithms more efficient in practice. The thesis also discusses the implications of these results for the parameterized complexity of other graph problems, and identifies several directions for future research. Overall, the thesis aims to better understand the practical implications of using twin-width in real-world scenarios while acknowledging its computational limitations.