On the plane angle-monotone graphs

Let P = ( p 1 , … , p n ) be a geometric path in the plane. For a real number 0 γ 180 ° , P is called an angle-monotone path with width γ if there exists a closed wedge of angle γ such that the vector of every edge ( p i , p i + 1 ) of P lies in this wedge. Let G be a geometric graph in the plane. The graph G is called angle-monotone with width γ if there exists an angle-monotone path with width γ between any two vertices. Let γ min be the smallest width such that every set of points in the plane has a plane angle-monotone graph with width γ min . It has been shown that 90 ° γ min ≤ 120 ° . In this paper, we show that γ min = 120 ° . This solves an open problem posed by Bonichon et al. [GD 2016].