Let $P=(p_1,\dots ,p_n)$ be a geometric path in the plane. For a real number $0<\gamma <180\mathrm{°}$, 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 ${\gamma }_{\min }$ be the smallest width such that every set of points in the plane has a plane angle-monotone graph with width ${\gamma }_{\min }$. It has been shown that $90\mathrm{°}<{\gamma }_{\min }\le 120\mathrm{°}$. In this paper, we show that ${\gamma }_{\min }=120\mathrm{°}$. This solves an open problem posed by Bonichon et al. [GD 2016].

让 $磷=({磷}_1,\dots ,{磷}_n)$是平面中的几何路径。对于实数$0<\gamma <180\mathrm{°}$, P称为宽度为 γ的角单调路径，如果存在角为γ的闭合楔形使得每条边的向量$({磷}_{\mathrm{一世}},{磷}_{\mathrm{一世}+1})$的P就在于此楔形。设G为平面上的几何图。该曲线图G ^称为角单调与宽度γ，如果存在与宽度的角度单调路径γ任意两个顶点之间。让${\gamma }_{\mathrm{分钟}}$ 是最小的宽度，使得平面中的每组点都有一个平面角单调图，宽度为 ${\gamma }_{\mathrm{分钟}}$. 已经表明$90\mathrm{°}<{\gamma }_{\mathrm{分钟}}\le 120\mathrm{°}$. 在本文中，我们表明${\gamma }_{\mathrm{分钟}}=120\mathrm{°}$. 这解决了 Bonichon 等人提出的一个开放问题。[广东 2016]。