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)

Notes on large angle crossing graphs

A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges in G intersect at an angle of at least a. The concept of right angle crossing (RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown that any RAC graph with n vertices has at most 4n-10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n-10 edges. In this paper, we give upper and lower bounds for the number of edges in aAC graphs for all 0 < a < Pi/2

On the Number of Edges of Fan-Crossing Free Graphs

A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties

The Effect of Planarization on Width

We study the effects of planarization (the construction of a planar diagram $D$ from a non-planar graph $G$ by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of $n$-vertex graphs with bounded parameter value, all of whose planarizations have parameter value $\Omega(n)$. However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Max-Cut and Max-Bisection are NP-hard on unit disk graphs

We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk graphs. We also show that $\lambda$-precision graphs are planar for $\lambda$ > 1 / \sqrt{2}$

Exact and fixed-parameter algorithms for metro-line crossing minimization problems

A metro-line crossing minimization problem is to draw multiple lines on an underlying graph that models stations and rail tracks so that the number of crossings of lines becomes minimum. It has several variations by adding restrictions on how lines are drawn. Among those, there is one with a restriction that line terminals have to be drawn at a verge of a station, and it is known to be NP-hard even when underlying graphs are paths. This paper studies the problem in this setting, and propose new exact algorithms. We first show that a problem to decide if lines can be drawn without crossings is solved in polynomial time, and propose a fast exponential algorithm to solve a crossing minimization problem. We then propose a fixed-parameter algorithm with respect to the multiplicity of lines, which implies that the problem is FPT.Comment: 19 pages, 15 figure