Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with $n$ vertices and $m$ edges can be computed in $2^m \cdot n^{O(1)}$ time. An $O(\log n)$-approximation of this number can be computed efficiently. We can decide in $2^{O(d \sqrt{k} \log (d+k))} \cdot n^{O(1)}$ time whether it suffices to delete $k$ edges of an ordered graph to obtain a $d$-planar layout (where every edge crosses at most $d$ other edges) on one page. As an additional parameter, we consider the size $h$ of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For $h=1$, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number $p$. For $h>1$, we give an XP algorithm with respect to $h+p$. Finally, we consider spine+$t$-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of $t$ tracks, each of which is a straight line on a separate page, parallel to the spine.

翻译：在平面上绘制图形并尽可能减少交叉数量是图绘制与计算几何领域的核心问题之一。另一种方法是删除最少数量的顶点或边，使得剩余图形可以无交叉地绘制。我们在有序图（即顶点顺序固定的图）的书嵌入设定中研究这两个问题。在该设定中，顶点按给定顺序位于一条称为书脊的直线上，每条边必须绘制在书的若干页之一，且每条边的交叉数量不超过固定上限。在书嵌入中，存在另一种减少或避免交叉的方式：使用更多页面。绘制有序图无需任何交叉所需的最少页面数被称为其（固定顶点顺序）页码。我们证明：具有$n$个顶点和$m$条边的有序图的页码可在$2^m \cdot n^{O(1)}$时间内计算，且该数值的$O(\log n)$近似值可高效求得。我们能在$2^{O(d \sqrt{k} \log (d+k))} \cdot n^{O(1)}$时间内判定：是否只需删除有序图中$k$条边，即可在单页上获得$d$平面布局（即每条边至多与其他$d$条边交叉）。作为额外参数，我们考虑命中集的大小$h$，即书脊上的一组点，使得每条边（视为开区间）至少包含其中一个点。当$h=1$时，我们可高效计算使固定顶点顺序页码降为$p$所需删除的最少边数。对于$h>1$，我们给出关于$h+p$的XP算法。最后，我们考虑脊线+$t$轨绘制：部分顶点位于书脊，而非全部顶点。书脊上的顶点顺序已给定；我们必须将每个不位于书脊的顶点映射到$t$条轨道之一，每条轨道是单独一页上平行于书脊的直线。