  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).
