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

For For 0 180+-,+, a geometric path P = (p1; : : : ; pn) is called angle-monotone with width from p1 to pn if there exists a closed wedge of angle such that every directed edge pipi+ of P lies inside the wedge whose apex is pi. A geometric graph G is called angle-monotone with width if for any two vertices p and q in G, there exists an anglemonotone path with width from p to q. In this paper, we show that for any integer k 1 and any i 2 f2; 3; 4; 5g, the theta-graph 4k+i on a set of points in convex position is anglemonotone with width 90 + i+ 4 , where + = 360+ 4k+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 0 180+, the graph 4 is not angle-monotone with width . Furthermore, we improve the stretch factor of graphs 4; 5; 7; 9; 11; Y4, and Y5, when the points are in convex position. Finally, we provide a lower bound of 3.66 for Y4 that solves an open problem.