Edge-minimum saturated k-planar drawings

For a class (Formula presented.) of drawings of loopless (multi-)graphs in the plane, a drawing (Formula presented.) is saturated when the addition of any edge to (Formula presented.) results in (Formula presented.) —this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on (Formula presented.) -planar drawings, that is, graphs drawn in the plane where each edge is crossed at most (Formula presented.) times, and the classes (Formula presented.) of all (Formula presented.) -planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated (Formula presented.) -planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all (Formula presented.) -vertex saturated (Formula presented.) -planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest (Formula presented.) -vertex saturated (Formula presented.) -planar drawings have (Formula presented.) edges for any (Formula presented.), while if all that is forbidden, the sparsest such drawings have (Formula presented.) edges for any (Formula presented.).