Edge‐minimum saturated k‐planar drawings - Chaplick, Steven; Klute, Fabian; Parada, Irene; Rollin, Jonathan; Ueckerdt, Torsten (Journal of graph theory, 2024)
For a class D of drawings of loopless (multi‐)graphs in the plane, a drawing D ∈ D is saturated when the addition of any edge to D results in D ′ ∉ D—this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on k‐planar drawings, that is, graphs drawn in the plane where each edge is crossed at most k times, and the classes D of all k‐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 k‐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 n‐vertex saturated k‐planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest n‐vertex saturated k‐planar drawings have 2 k − ( k mod 2 ) ( n − 1 ) edges for any k ≥ 4, while if all that is forbidden, the sparsest such drawings have 2 ( k + 1 ) k ( k − 1 ) ( n − 1 ) edges for any k ≥ 6.