A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a unit length vertical segment, and one wants to connect pairs of segments with...

Non-crossing knight‟s tour in 3-dimension is a new field of research. The author has shown its possibility in small cuboids and in cubes up to 8x8x8 size. It can also be extended to larger size cubes and cuboids. The author has achieved jumps of length 15, 46, 88, 159, 258 and 395 in cubes of size 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7 and 8x8x8 respectively. This amounts to covering 59%, 73%, 71%, ...

We consider the problem of on-line scheduling with non-crossing constraints. The objective is to minimize the latest completion time. We provide optimal competitive ratio heuristics for the online-list and online-time problems with unit processing times, and a 3-competitive heuristic for the general online-time problem.

Let P be a set of 2n points in the plane, and letMC (resp.,MNC) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of P . We study the problem of computing MNC. We first prove that the problem is NP-hard and does not admit a PTAS. Then, we present an O(n log n)-time algorithm that computes a noncrossing matching M of P , such that bn(M) ≤ 2 √ 10 · bn(MNC), where bn(M) is t...