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We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph G = (V, E) is (k, p)-planar if V can be partitioned into clusters of size at most k such that G admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region; (ii) cluster regions are pairwise disjoint, (iii) each vertex v ∈ V is identified with at most p distinct points, called ports, on the boundary of its cluster region; (iv) each inter-cluster edge (u, v) ∈ E is identified with a Jordan arc connecting a port of u to a port of v; (v) inter-cluster edges do not cross or intersect cluster regions except at their endpoints. We first tightly bound the number of edges in a (k, p)-planar graph with p < k. We then prove that (4, 1)-planarity testing and (2, 2)-planarity testing are NP-complete problems. Finally, we prove that neither the class of (2, 2)-planar graphs nor the class of 1planar graphs contains the other, indicating that the (k, p)-planar graphs are a large and novel class.

We study the question whether a crossing-free 3D morph between two straight-line drawings of an n-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with O(log n) steps, while for the latter Θ(n) steps are always sufficient and sometimes necessary. * We here refer to pole dancing as a fitness and competitive sport. The authors hope that many of our readers try this activity themselves, and will in return introduce many pole dancers to Graph Drawing, thereby alleviating the gender imbalance in both communities. The authors do not condone any pole activity used for sexual exploitation or abuse of women or men. between any two topologically-equivalent † planar straight-line ‡ drawings of the same planar graph always exists; this was proved for maximal planar graphs by Cairns [8] back in 1944, and then for all planar graphs by Thomassen [16] almost forty years later. Note that a planar morph between two planar graph drawings that are not topologically equivalent does not exist.It has lately been well investigated whether a planar morph between any two topologically-equivalent planar straight-line drawings of the same planar graph always exists such that the vertex trajectories have low complexity. This is usually formalized as follows. Let Γ and Γ be two topologically-equivalent planar straight-line drawings of the same planar graph G. Then a morph M is a sequence Γ 1 , Γ 2 , . . . , Γ k of planar straight-line drawings of G such that Γ 1 = Γ , Γ k = Γ , and Γ i , Γ i+1 is a planar linear morph, for each i = 1, . . . , k − 1. A linear morph Γ i , Γ i+1 is such that each vertex moves along a straight-line segment at uniform speed; that is, assuming that the morph happens between time t = 0 and time t = 1, the position of a vertex v at any time t ∈The complexity of a morph M is then measured by the number of its steps, i.e., by the number of linear morphs it consists of.A recent sequence of papers [3,4,5,6] culminated in a proof [2] that a planar morph between any two topologically-equivalent planar straight-line drawings of the same n-vertex planar graph can always be constructed consisting of Θ(n) steps. This bound is asymptotically optimal in the worst case, even for paths.The question we study in this paper is whether morphs with sub-linear complexity can be constructed if a third dimension is allowed to be used. That is: Given two topologically-equivalent planar straight-line drawings Γ and Γ of the same n-vertex planar graph G does a morph M = Γ = Γ 1 , Γ 2 , . . . , Γ k = Γ exist such that: (i) for i = 1, . . . , k, the drawing Γ i is a crossing-free straight-line 3D drawing of G, i.e., a straight-line drawing of G in R 3 such that no two edges cross; (ii) for i = 1, . . . , k − 1, the step Γ i , Γ i+1 is a crossing-free linear morph, i.e., no two edges cross throughout the tr...

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