Edge-Minimum Saturated k-Planar Drawings

Abstract

For a class D of drawings of loopless multigraphs 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. For k ≥ 4, 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 ≥ 7.

Saturated 2-plane drawings with few edges

“…There are several versions of these concepts, see e.g. [4]. The most studied one is when we consider only simple drawings.…”

“…The saturation problem for k-planar graphs has many different settings, we can allow self-crossings, parallel edges, or we can consider non-extendable abstract graphs. See [4] for many recent results and a survey.…”