The crossing number of a graph $G$ is the minimum number of crossings in a drawing of $G$ in the plane. A rectilinear drawing of a graph $G$ represents vertices of $G$ by a set of points in the plane and represents each edge of $G$ by a straight-line segment connecting its two endpoints. The rectilinear crossing number of $G$ is the minimum number of crossings in a rectilinear drawing of $G$. By the crossing lemma, the crossing number of an $n$-vertex graph $G$ can be $O(n)$ only if $|E(G)|\in O(n)$. Graphs of bounded genus and bounded degree (B\"{o}r\"{o}czky, Pach and T\'{o}th, 2006) and in fact all bounded degree proper minor-closed families (Wood and Telle, 2007) have been shown to admit linear crossing number, with tight $\Theta(\Delta n)$ bound shown by Dujmovi\'c, Kawarabayashi, Mohar and Wood, 2008. Much less is known about rectilinear crossing number. It is not bounded by any function of the crossing number. We prove that graphs that exclude a single-crossing graph as a minor have the rectilinear crossing number $O(\Delta n)$. This dependence on $n$ and $\Delta$ is best possible. A single-crossing graph is a graph whose crossing number is at most one. Thus the result applies to $K_5$-minor-free graphs, for example. It also applies to bounded treewidth graphs, since each family of bounded treewidth graphs excludes some fixed planar graph as a minor. Prior to our work, the only bounded degree minor-closed families known to have linear rectilinear crossing number were bounded degree graphs of bounded treewidth (Wood and Telle, 2007), as well as, bounded degree $K_{3,3}$-minor-free graphs (Dujmovi\'c, Kawarabayashi, Mohar and Wood, 2008). In the case of bounded treewidth graphs, our $O(\Delta n)$ result is again tight and improves on the previous best known bound of $O(\Delta^2 n)$ by Wood and Telle, 2007 (obtained for convex geometric drawings).

翻译：图$G$的交叉数是指在平面绘制$G$时交叉的最小数量。图$G$的直线绘制将$G$的顶点表示为平面点集，并将每条边表示为连接其两端点的直线段。$G$的直线交叉数是在$G$的直线绘制中交叉的最小数量。根据交叉引理，一个$n$顶点图$G$的交叉数可能为$O(n)$，仅当$|E(G)|\in O(n)$。有界亏格和有界度图（Böröczky, Pach 和 Tóth, 2006），实际上所有有界度的真子式闭族（Wood 和 Telle, 2007）已被证明具有线性交叉数，且由 Dujmović, Kawarabayashi, Mohar 和 Wood (2008) 给出紧的 $\Theta(\Delta n)$ 界。关于直线交叉数的研究则少得多。它不受任何交叉数函数的界限。我们证明，排除单交图作为子式的图具有直线交叉数 $O(\Delta n)$。这一对$n$和$\Delta$的依赖性是最优的。单交图是指交叉数至多为1的图。因此，该结果例如适用于不含$K_5$子式的图。它也适用于有界树宽图，因为每个有界树宽图族都排除某个固定平面图作为子式。在我们的工作之前，已知具有线性直线交叉数的有界度子式闭族仅有有界树宽的有界度图（Wood 和 Telle, 2007），以及有界度的不含$K_{3,3}$子式的图（Dujmović, Kawarabayashi, Mohar 和 Wood, 2008）。对于有界树宽图，我们的 $O(\Delta n)$ 结果同样是紧的，并改进了 Wood 和 Telle (2007) 之前最佳已知的 $O(\Delta^2 n)$ 界（该结果针对凸几何绘制获得）。