1 research outputs found
Edge-Minimum Saturated k-Planar Drawings
For a class of drawings of loopless multigraphs in the plane, a
drawing is saturated when the addition of any edge to
results in . 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 -planar drawings, that is, graphs drawn in the plane
where each edge is crossed at most times, and the classes of
all -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 -planar drawings are the focus of several prior works,
tight bounds on how sparse these can be are not well understood. For , we establish a generic framework to determine the minimum number of edges
among all -vertex saturated -planar drawings in many natural classes. For
example, when incident crossings, multicrossings and selfcrossings are all
allowed, the sparsest -vertex saturated -planar drawings have edges for any , while if all that is forbidden,
the sparsest such drawings have edges for any .Comment: Added a paragraph commenting on recent independent work by Klute and
Parada (arXiv:2012.02281