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    Edge-Minimum Saturated k-Planar Drawings

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    For a class D\mathcal{D} of drawings of loopless multigraphs in the plane, a drawing D∈DD \in \mathcal{D} is saturated when the addition of any edge to DD results in Dβ€²βˆ‰DD' \notin \mathcal{D}. This is analogous to saturated graphs in a graph class as introduced by Tur\'an (1941) and Erd\H{o}s, Hajnal, and Moon (1964). We focus on kk-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most kk times, and the classes D\mathcal{D} of all kk-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 kk-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. For kβ‰₯4k \geq 4, we establish a generic framework to determine the minimum number of edges among all nn-vertex saturated kk-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest nn-vertex saturated kk-planar drawings have 2kβˆ’(kβ€Šmodβ€Š2)(nβˆ’1)\frac{2}{k - (k \bmod 2)} (n-1) edges for any kβ‰₯4k \geq 4, while if all that is forbidden, the sparsest such drawings have 2(k+1)k(kβˆ’1)(nβˆ’1)\frac{2(k+1)}{k(k-1)}(n-1) edges for any kβ‰₯7k \geq 7.Comment: Added a paragraph commenting on recent independent work by Klute and Parada (arXiv:2012.02281
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