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Titel:

Edge-minimum saturated k-planar drawings

AutorInnen:

Steven Chaplick
Fabian Klute
Irene Parada
Jonathan Rollin
Torsten Ueckerdt

Kategorie:

Artikel in Zeitschriften

erschienen in:

Journal of Graph Theory, Vol. 106(4), pp. 741-762, 2024

Abstract:

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 (1941) and Erdős, Hajnal, and Moon (1964). 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 .

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