Recognizing Weakly Simple Polygons
  We present an $O(n\log n)$-time algorithm that determines whether a given planar $n$-gon is weakly simple. This improves upon an $O(n^2\log n)$-time algorithm by Chang, Erickson, and Xu (2015). Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question. 
Hugo Akitaya, Greg Aloupis, Jeff Erickson, Csaba D. T\'oth
cs.CG