Angle-monotonicity of theta-graphs for points in convex position

For a real number 0 < γ < 180 ◦ , a geometric path P = ( p 1 , . . . , p n ) is called angle-monotone with width γ from p 1 to p n if there exists a closed wedge of angle γ such that every directed edge −−−−→ p i p i +1 of P lies inside the wedge whose apex is p i . A geometric graph G is called angle-monotone with width γ if for any two vertices p and q in G , there exists an angle-monotone path with width γ from p to q . In this paper, we show that for any integer k ≥ 1 and any i ∈ { 2 , 3 , 4 , 5 } , the theta-graph Θ 4 k + i on a set of points in convex position is angle-monotone with width 90 ◦ + iθ 4 , where θ = 360 ◦ 4 k + i . Moreover, we present two sets of points in the plane, one in convex position and the other in non-convex position, to show that for every 0 < γ < 180 ◦ , the graph Θ 4 is not angle-monotone with width γ . ,