Plane Multigraphs with One-Bend and Circular-Arc Edges of a Fixed Angle

For an angle \(\alpha \in (0,\pi )\) , we consider plane graphs and multigraphs in which the edges are either (i) one-bend polylines with an angle \(\alpha \) between the two edge segments, or (ii) circular arcs of central angle \(2(\pi -\alpha )\) . We derive upper and lower bounds on the maximum density of such graphs in terms of \(\alpha \) . As an application, we improve upon bounds for the number of edges in \(\alpha AC_1^=\) graphs (i.e., graphs that can be drawn in the plane with one-bend edges such that any two crossing edges meet at angle \(\alpha \) ). This is the first improvement on the size of \(\alpha AC_1^=\) graphs in over a decade.