Convex geometric (k + 2) -quasiplanar representations of semi-bar k-visibility graphs

Semi-bar k-visibility graphs are graphs for which vertices can be drawn as horizontal segments with left endpoints on the y-axis (semi-bars) and edges can be drawn as vertical segments (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. Cyclic semi-bar k-visibility graphs are graphs for which vertices can be drawn as segments with one endpoint at the origin (semi-bars) and edges can be drawn as arcs of circles centered at the origin (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. We show that every semi-bar or cyclic semi-bar k-visibility graph can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k+2 pairwise crossing edges. Furthermore, we prove that the class of graphs having cyclic semi-bar k-visibility representations with semi-bars of different lengths is the same as the class of (2k+2)-degenerate graphs that can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k+2 pairwise crossing edges and so that adding any edge would result in k+2 pairwise crossing edges.