String graphs and separators

Abstract: 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.