COVID crossing

CoViD, A System for Collaborative Virtual 3D Design

Many important decisions in the design process are made fairly early on, after designers have presented initial concepts. In many domains, these concepts are now realized as 3D digital models and, in meetings, the stakeholders evaluate these potential solutions. Frequently, the participants in such a meeting want to interactively modify the proposed designs to explore the design space better. T...

Crossing technique in cowpeas

Non-Crossing Frameworks with Non-Crossing Reciprocals

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks G whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the stress on G; and a geometric condition on ...

Odd Crossing Number Is Not Crossing Number

The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbe...

Crossing number, pair-crossing number, and expansion

The crossing number crðGÞ of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane, while the pair-crossing number pcrðGÞ is the smallest number of pairs of edges that cross in a drawing of G in the plane. While crðGÞXpcrðGÞ holds trivially, it is not known whether a strict inequality can ever occur (this question was raised by Mohar and Pach and Tóth). We ai...