Over the past decade, we witness an increasing amount of interest in the design of exact exponential-time and parameterized algorithms for problems in Graph Drawing. Unfortunately, we still lack knowledge of general methods to develop such algorithms. An even more serious issue is that, here, "standard" parameters very often yield intractability. In particular, for the most common structural parameter, namely, treewidth, we frequently observe NP-hardness already when the input graphs are restricted to have constant (often, being just $1$ or $2$) treewidth. Our work deals with both drawbacks simultaneously. We introduce a novel form of tree decomposition that, roughly speaking, does not decompose (only) a graph, but an entire drawing. As such, its bags and separators are of geometric (rather than only combinatorial) nature. While the corresponding parameter -- like treewidth -- can be arbitrarily smaller than the height (and width) of the drawing, we show that -- unlike treewidth -- it gives rise to efficient algorithms. Specifically, we get slice-wise polynomial (XP) time algorithms parameterized by our parameter. We present a general scheme for the design of such algorithms, and apply it to several central problems in Graph Drawing, including the recognition of grid graphs, minimization of crossings and bends, and compaction. Other than for the class of problems we discussed in the paper, we believe that our decomposition and scheme are of independent interest and can be further extended or generalized to suit even a wider class of problems. Additionally, we discuss classes of drawings where our parameter is bounded by $O(\sqrt{n})$ (where $n$ is the number of vertices of the graph), yielding subexponential-time algorithms. Lastly, we prove which relations exist between drawn treewidth and other width measures, including treewidth, pathwidth, (dual) carving-width and embedded-width.

翻译：在过去十年中，我们见证了针对图绘制问题的精确指数时间算法和参数化算法设计日益浓厚的兴趣。遗憾的是，我们仍缺乏开发此类算法的通用方法。一个更为严重的问题是，在此领域中"标准"参数往往导致难解性。具体而言，对于最常见的结构参数即树宽，我们经常观察到当输入图被限制为具有常数（通常仅为1或2）树宽时，问题已然是NP难的。我们的工作同时解决了这两个缺陷。我们引入了一种新颖的树分解形式，大致而言，它不仅分解（仅仅）一个图，而是分解整个绘制。因此，其袋子和分离子具有几何（而非仅组合）性质。虽然相应参数（如同树宽）可能任意小于绘制高度（和宽度），但我们证明——与树宽不同——该参数可以催生高效算法。具体而言，我们得到了以该参数为参数的切片多项式（XP）时间算法。我们提出了此类算法设计的通用方案，并将其应用于图绘制中的若干核心问题，包括网格图识别、交叉点和弯路最小化以及紧凑化。除了论文中讨论的问题类别之外，我们相信我们的分解和方案具有独立价值，并可进一步扩展或泛化以适应更广泛的问题类别。此外，我们讨论了若干绘制类别，其中我们的参数被限制为$O(\sqrt{n})$（其中$n$是图的顶点数），从而得到次指数时间算法。最后，我们证明了绘制树宽与其他宽度度量（包括树宽、路径宽、（对偶）雕刻宽和嵌入宽）之间存在的关系。