String graphs and separators

String graphs, that is, intersection graphs of curves in the plane, have been studied since the 1960s. We provide an expository presentation of several results, including very recent ones: some string graphs require an exponential number of crossings in every string representation; exponential number is always sufficient; string graphs have small separators; and the current best bound on the crossing number of a graph in terms of the pair-crossing number. For the existence of small separators, unwrapping the complete proof include generally useful results on approximate flow-cut dualities.Comment: Expository paper based on course note

The Crossing Number of Two Cartesian Products

There are several known exact results on the crossing number of Cartesian products of paths, cycles, and complete graphs. In this paper, we find the crossing numbers of Cartesian products of Pn with two special 6-vertex graphs