Link Crossing Number is NP-hard

分裂链环的困难图

Hard diagrams of split links

研究将分裂链环图转变为分裂图的问题，利用气泡缠结框架及特定技术，证明存在分裂链环，其图转变为分裂图时需任意大量额外交叉。

Resolution-Exact Planner for Thick Non-Crossing 2-Link Robots

We consider the path planning problem for a 2-link robot amidst polygonal obstacles. Our robot is parametrizable by the lengths $\ell_1, \ell_2>0$ of its two links, the thickness $τ\ge 0$ of the links, and an angle $κ$ that constrains the angle between the 2 links to be strictly greater than $κ$. The case $τ>0$ and $κ\ge 0$ corresponds to "thick non-crossing" robots. This results in a novel 4DOF configuration space ${\mathbb R}^2\times ({\mathbb T}\setminusΔ(κ))$ where ${\mathbb T}$ is the torus and $Δ(κ)$ the diagonal band of width $κ$. We design a resolution-exact planner for this robot usin…

Hardness of approximation for crossing number

We show that, if P\not=NP, there is a constant c > 1 such that there is no c-approximation algorithm for the crossing number, even when restricted to 3-regular graphs.

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-…

路径宽度为常数的交叉数问题是NP难的

Crossing Number is NP-hard for Constant Path-width (and Tree-width)

本文证明了在路径宽度为12的图中计算交叉数仍为NP难问题，挑战了传统图分解在解决交叉数问题上的有效性。

Adding one edge to planar graphs makes crossing number and 1-planarity hard

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which i…