Graph drawing problems originate from diverse application domains. In some, such as software engineering and cartography, graphs are required to be visualized or drawn in ways that are easy to read and understand. In others, such as VLSI design, graphs are required to be laid out while satisfying some physical constraint. For example, when a drawing is to be displayed on a page or a computer screen, or is to be used for VLSI design, it is important to keep its area/volume small to avoid wasting space. More often than not however, the idea of a good drawing, regardless of its purpose, coincides with having no edge crossings or having very few crossings. Unfortunately, whichever of the numerous drawing styles one considers, a problem requiring a crossing minimization of sorts will, almost certainly, be NP -hard. The theory of fixed parameter tractability (FPT) provides a new and promising approach for coping with intractable problems. In the first part of this thesis we apply algorithmic techniques developed in this theory to well-known graph drawing problems. In particular, we contribute efficient FPT algorithms for crossing minimization and planarization problems concerning the 2-layer drawing style. In the second part of this thesis we introduce and comprehensively study so-called track layouts of graphs and their subdivisions. A relationship between this combinatorial structure and several well-known types of graph layouts is established, leading to a number of new results. For example, our study of track layouts of bounded treewidth graphs settles an open problem due to Ganley and Heath (2001) regarding queue layouts of such graphs. Moreover, the study also establishes that graphs of bounded treewidth have three-dimensional straight-line grid drawings with linear volume. Through the study of track layouts of subdivisions, we determine that every graph with n vertices and m edges has a three-dimensional polyline grid drawing with the vertices on a rectangular prism, O (n + m log n) volume and O (log n) bends per edge.

Whitesides, Sue (Supervisor)