Summary: For a class \(\mathcal{D}\) of drawings of loopless (multi-)graphs in the plane, a drawing \(D \in \mathcal{D}\) is saturated when the addition of any edge to \(D\) results in \(D^\prime \notin \mathcal{D} \) – this is analogous to saturated graphs in a graph class as introduced by P. Turán [Mat. Fiz. Lapok 48, 436–452 (1941; Zbl 0026.26903; JFM 67.0732.03)] and P. Erdős et al. [Am. Math. Mon. 71, 1107–1110 (1964; Zbl 0126.39401)]. 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 \(\mathcal{D}\) 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 \(\frac{2}{k - ( k \bmod 2 )} ( n - 1 )\) edges for any \(k \geq 4\), while if all that is forbidden, the sparsest such drawings have \(\frac{2 ( k + 1 )}{k ( k - 1 )} ( n - 1 )\) edges for any \(k \geq 6\).
© 2024 The Authors. Journal of Graph Theory published by Wiley Periodicals LLC.