On the Relationship between $k$-Planar and $k$-Quasi Planar Graphs

Two-Planar Graphs Are Quasiplanar

“…A 3-crossing in a drawing is untangled if the six endpoints of the edges lie on the same face of the arrangement formed by the three edges; otherwise the 3-crossing is tangled, see Figure 2a-2b for an example. Angelini et al showed [6,Lemma 2] that every 2-planar graph admits a 2-plane drawing in which every 3-crossing is untangled. Their proof starts from a 2plane drawing and rearranges tangled 3-crossings without introducing any new edge crossings.…”

“…2 Angelini et al proved [6,Lemma 3 and 4] that there exists an injective map f : H → V that maps every hexagon h ∈ H to a vertex v ∈ V(h). For each hexagon h ∈ H, exactly one edge in h is incident to the vertex f (h).…”

“…The relation between k-planarity and -quasiplanarity has been studied only recently. Angelini et al [6] proved that for k ≥ 3, every k-planar graph is (k + 1)-quasiplanar. However, the case k = 2 was left open.…”

“…Angelini et al [6] proved that for k ≥ 3, every k-planar graph admits a (k + 1)-quasiplane simple topological drawing. A careful analysis of our redrawing algorithm, which transforms a 2-plane drawing of a graph into a quasiplane drawing, reveals that it produces a quasiplane simple topological drawing.…”

“…The concept of k-planarity plays a crucial role in proving the current best constants for the classic Crossing Lemma [2,5,17], and k-quasiplanarity is closely related to Ramsey-type properties of the intersection graph of Jordan arcs in the plane [4]. However, relations between the latter two graph classes have been studied only recently [6]. k-planarity.…”