Rectilinear Crossing Number of Graphs Excluding Single-Crossing Graphs as Minors

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).