Computing a morph between two drawings of the same graph is a classical problem that attracted considerable attention over the years, also in view of its numerous applications in computer graphics and animations (refer to  for a short overview). At high level, given two drawings and of the same graph , a morph between and is a continuously changing family of drawings such that the initial one coincides with and the final one with . A standard assumption is that the two input drawings - as well as all intermediate ones - are topologically equivalent, i.e., they define the same set of cells up to a homeomorphism of the plane (see Section 2 for formal definitions). The main challenge is to design morphing algorithms that maintain some additional geometric properties of the input drawings throughout the transformation, such as planarity with straight-line edges (see, e.g., [1, 16, 24]), convexity [6, 33], orthogonality [12, 25, 25], and upwardness . We point the interested reader to [4, 8, 10, 11] for additional related work.
In this context, the most prominent research direction focuses on morphs of straight-line planar drawings: The topological equivalence condition implies that all drawings in the morph have the same planar embedding; in addition, it is also required that edges remain straight-line segments. Back in 1944, Cairns  proved that such morphs always exist if the input graphs are plane triangulations. This implies that, for a fixed plane triangulation, the space of its straight-line planar drawings is connected. The main drawback of Cairns result is in the underlying construction, which involves exponentially-many morphing steps. The extension of Cairns’ result to all plane graphs was initially done by Thomassen , while later Floater and Gotsman , and Gotsman and Surazhsky [26, 31] proposed different approaches using trajectories of unbounded complexity. More recently, Alamdari et al.  focused on the complexity of the morph. They described the first morphing algorithm for planar straight-line drawings that makes use of a polynomial number of steps, where in each step vertices move at uniform speed along linear trajectories. In a subsequent paper , a linear bound on the number of steps is shown, which is worst-case optimal.
Morphing non-planar drawings of graphs appears to be a more elusive problem. In particular, Angelini et al.  asked whether a morphing algorithm exists for pairs of non-planar straight-line drawings such that the topology of the crossings in the drawing is maintained throughout the morph. They stressed that a solution to this problem is not known even if the vertex trajectories are allowed to have arbitrary complexity. Note that the obvious idea of morphing the “planarizations” of the drawings (i.e., the planar drawings obtained by treating crossings as dummy vertices) does not trivially work. Namely, in order to guarantee that edges remain straight-line segments throughout the morph, one has to ensure that opposite edges incident to dummy vertices maintain the same slope. To the best of our knowledge, such requirement cannot be easily incorporated into any of the already known morphing algorithms for planar graphs.
One way of simplifying the problem is to consider graphs that are non-planar but still admit embeddings on surfaces of bounded genus. In this direction, Chambers et al.  proved the existence of morphs for pairs of crossing-free drawings on the Euclidean flat torus (where edges are still geodesics). Their technique is rather complex and the authors concluded that an extension to higher genus surfaces is fairly non-trivial.
We make a step towards settling the open problem in  by studying non-planar drawings of graphs with forbidden edge-crossing patterns. Our focus is on the family of -planar graphs, which naturally extends the notion of planarity by allowing each edge to be crossed at most once (see  for a survey). Note that -planar graphs form a well studied family of non-planar graphs with early results dating back to the 60’s [9, 30], while more recently they have gained considerable attention in the rapidly growing literature about beyond planarity [23, 27].
Our contribution. We provide a set of sufficient conditions under which any pair of -planar straight-line drawings admits a morph. At high-level, we require that if two edges cross, then they can be enclosed in a quadrilateral region whose boundary is uncrossed; although this region may contain further vertices in its interior, we require that any edge connecting an end-vertex of the crossing edges to a vertex inside the region is also uncrossed; refer to Fig. 1 for an illustration. A drawing that satisfies these requirements is called kite-planar -planar (see Definition 1). Our main result is summarized by the following theorem.
There exists a morph between any pair of topologically-equivalent kite-planar -planar drawings.
Theorem 1.1 implies that, for a fixed graph, the space of its topologically-equivalent kite-planar -planar drawings is connected. The proof is constructive, although the vertices may use trajectories of unbounded complexity. Concerning the definition of kite-planar -planar drawings, it may be worth observing that, due to a simple edge density argument, the graphs admitting such a drawing cannot be embedded on any surface of bounded genus. Indeed, as shown in Section 6, some well-known families of -planar graphs admit drawings that are kite-planar -planar and require arbitrary large genus to be embedded.
Paper structure. Section 2 contains basic definitions and notation. Section 3 gives an overview of the proof technique, which exploits a recursive construction. The base case of the recursion is described in Section 4, while the recursive step is in Section 5. Implications of our result in terms of classes of -planar drawings that admit a morph are discussed in Section 6. Open problems are given in Section 7.
Drawings. A straight-line drawing (or simply a drawing, for short) of a graph maps each vertex of to a distinct point of the plane and each edge of to a straight-line segment connecting and without passing through any other point representing a vertex of . When this creates no ambiguities, we will not distinguish between a vertex and the point representing it in , as well as between an edge and its segment. Note that, by definition, two edges of a drawing share at most one point, which is either a common endpoint or an interior point where the two edges properly cross. Drawing partitions the plane into connected regions called cells. The boundary of a cell consists of vertices, crossing points, and (parts of) edges. The external cell of is its (only) unbounded cell. Two drawings and of the same graph are topologically equivalent if they define the same set of cells up to a homeomorphism of the plane. An embedding of is an equivalence class of drawings that are pairwise topologically equivalent.
A drawing is planar if no two edges cross. In this case, the cells of are called faces and their boundaries consist of just vertices and edges. A graph is planar if it admits a planar drawing. A planar graph together with an embedding defined by a planar drawing is a plane graph. A planar drawing is strictly convex if all its faces are strictly convex polygons.
A graph is -planar if it admits a (not necessarily straight-line) -planar drawing in which every edge crosses at most one other edge. A -planar graph together with an embedding defined by a -planar drawing is a -plane graph. A kite in a -planar drawing is a -planar drawing of in whose external cell is a quadrilateral. The four edges on the boundary of the external cell of are called kite edges. The other two edges are the crossing edges of and are drawn inside the quadrilateral bounding . A partial kite is a subdrawing of a kite such that some kite edges may not exist. Figure 1 shows two kites and one partial kite; the kite (crossing) edges are blue (red, resp.).
Throughout the paper, we assume that the four kite-edges of any (partial) kite exist and are uncrossed; this will be justified by P.2 of Definition 1. In other words, we assume that, in the considered drawings, no kite is partial. Given a vertex of and a kite , the following exclusive cases can occur: (i) belongsto , if it is a vertex of the defining , or (ii) is inside (or contains ) if lies in the interior of the quadrilateral bounding , or (iii) is outside , otherwise. A kite is empty if it contains no vertex; otherwise, it is non-empty. An edge is a binding edge (green in Fig. 1) if belongs to a non-empty kite and is inside . We can now introduce kite-planar -planar drawings.
A straight-line drawing is kite-planar -planar, or -kite-planar for short, if: (P.1) every edge is crossed at most once, (P.2) the four kite edges of every (possibly partial) kite are either present or can be drawn with uncrossed straight-line segments, and (P.3) every binding edge is uncrossed.
Let be a -kite-planar drawing of . We say that a vertex of is of level if no kite contains it, while it is of level if the maximum level of the vertices belonging to a kite containing it is . In Fig. 1, the black (white) vertices are of level (level , resp.). The next property follows from P.3 of Definition 1.
If two vertices belong to the same kite of a -kite-planar drawing , then they are of the same level.
Suppose, for a contradiction, that two vertices and that belong to a kite are such that the level of is larger than the level of . This implies that is inside a kite that does not contain . By 1-planarity, is a kite-vertex of . It follows that there is a crossing edge of incident to that is a binding edge with respect to , which contradicts Property P.3 of Definition 1.
Morphs. Let and be two topologically-equivalent drawings of the same graph . A morph between them is a continuously changing family of pairwise topologically-equivalent drawings of indexed by time , such that the drawing at time is and the drawing at time is . Since edges are drawn as straight-line segments, a morph is uniquely specified by the vertex trajectories. Also, during the course of the morph, a vertex may coincide with neither another vertex nor an internal point of an edge.
3 Outline of the Proof of Theorem 1.1
In this section, we give an outline of the proof of Theorem 1.1, namely, that there exists a morph between any two topologically-equivalent -kite-planar drawings and of a graph . Recall that and define the same embedding of . Hence, is necessarily a -plane graph.
Our proof is by means of a recursive construction. The underlying idea is to compute a morph by keeping each kite boundary drawn as a strictly-convex polygon, so that, in the course of the morph, the drawing of the corresponding crossing edges will stay inside their boundary. The main challenge, however, stems from the fact that a kite may not be empty. Therefore, our approach is to remove the interior of each kite, recursively compute a morph that keeps the convexity of the kite boundaries, and suitably reinsert (and morph) the removed subdrawings. In the proof, we will use two key ingredients. The first one is a result by Aronov et al. , which guarantees that one can compatibly triangulate two topologically-equivalent planar drawings of a planar graph.
Theorem 3.1 (Aronov et al. )
Given two topologically-equivalent planar drawings and of the same -vertex planar graph , it is possible to augment and to two topologically-equivalent planar drawings and of the same maximal planar graph such that , , and the order of is .
The second ingredient is a result by Angelini et al. , which allows us to morph a pair of convex drawings by preserving the convexity of the faces. The main properties of this result are summarized in the next theorem.
Theorem 3.2 (Angelini et al. )
Let be a pair of topologically-equivalent strictly-convex planar drawings of a graph . There is a morph between and in which every intermediate drawing is strictly convex. If the outer face of has only three vertices and each of them has the same position in and , then these three vertices do not move during this morph.
4 Base case
In the base case of the recursion, all the vertices of are of level , which implies that all the kites of , if any, are empty. Let be the graph obtained by removing both crossing edges from each kite of . Let be the restrictions of to , respectively; see Fig. 2. By construction, is a pair of planar and topologically-equivalent drawings, and is a plane subgraph of . The kite edges of each kite of are uncrossed (by P.2 of Definition 1) and bound a quadrangular face in , which we call marked.
Let and be the graph and the corresponding pair of planar drawings obtained by applying Theorem 3.1 to , except for the marked faces; see Fig. 2 for an illustration. This operation guarantees that every face in both drawings is a triangle, if not marked, or a quadrangle, if marked. We call a plane graph with such faces almost triangulated, and we next prove that it is triconnected.
Every almost triangulated plane graph is triconnected.
Let be an almost triangulated plane graph derived from a -kite-planar drawing of a -planar graph . Suppose that contains a separation pair . Then there exist at least two faces and that are incident to both and such that at least one, say , is not triangular and hence is marked. In the presence of this marked face, edge exists in and not in . Consequently, cannot be a triangle, as otherwise it would contain edge on its boundary. On the other hand, if is marked, then contains another copy of drawn inside the kite that yielded , which is impossible since is simple. Hence, contains no separation pair. The absence of cutvertices stems from the fact that each face is either a triangle or a quadrangle (if marked).
Since each kite contains two crossing edges in , its boundary is drawn strictly convex in both and . Hence, and are two strictly convex planar drawings of . This property allows to apply Theorem 3.2 to compute a morph of that maintains the strict convexity of the drawing at any time instant. Since each marked face remains strictly convex, adding back the two crossing edges of the corresponding kite in yields a morph of a supergraph of (and thus of ) in which these crossing edges remain inside the boundary of at any time instant. This concludes the base case.
5 Recursive case
In this section, we focus on the recursive step of the proof of Theorem 1.1, in which the maximum level of a vertex in is . Let be the graph obtained by removing all the vertices of level from , and let be the restriction of to . Clearly, the two drawings of are topologically equivalent and the maximum level of a vertex is . Thus, we can recursively compute a morph of . In what follows, we describe how to incorporate the trajectories of the level- vertices into the morph of , so to obtain the desired morph of .
Setting up the morph. We begin by observing that, by creftypecap 1, each vertex of level is contained in a kite whose vertices are all of level , which implies that this kite is empty in (but not in ). Consider such a kite , and note that its two crossing edges define four triangular regions that remain non-degenerate during the morph of . We refer to each of these four triangular regions as a piece of a kite. Consider a piece of kite and denote it by . The unique edge of that belongs to the boundary of is called the base edge of . Since remains non-degenerate during the morph, there exists a half-disk that, throughout the whole morph, has the following properties (see also Fig. 4 for an illustration):
half-disk lies in and is centered at the midpoint of , and
the length of its radius is positive and it does not change.
Let be the smallest length of the base edge during the morph, let be the radius of perpendicular to , and let be the endpoint of different from . Also, denote by any time instant of the morph when the length of equals , and let be the internal angle at of the triangle formed by and at time . In particular, satisfies .
Consider the graph induced by the level- vertices of , and let be the subgraph of that lies inside . We proceed to compute a drawing of that, intuitively, will be “small” enough to fit inside and “skinny” enough to avoid crossings with the binding edges that connect or to . To ease the notation, from now on we will refer to as .
To compute this drawing, we first augment as well as its drawings in , as follows. We add a dummy vertex connected to and to , which is drawn sufficiently close to the crossing point of the two diagonals of in both and , so that the triangle formed by , , and contains .
As in the transition from to in Section 4, we remove the crossing edges of every (empty) kite of and we mark the resulting quadrangular face. Then we apply Theorem 3.1 to the resulting planar subgraph of and to its drawings in , except for its marked faces. This results in an almost triangulated plane graph and in a pair of topologically-equivalent strictly convex drawings of . The following observation is directly implied by Property P.3 of Definition 1.
Every face incident to or to in is triangular.
Consider the plane graph obtained from by removing and , and let be the graph formed by the vertices and the edges of its outer face. In the following lemma, we investigate some properties of . The BC-tree of a connected graph represents the decomposition of into its biconnected components, called blocks. Namely, has a B-node for each block of and a C-node for each cutvertex of , such that each B-node is connected to the C-nodes that are part of its block.
The following properties of hold:
is outerplane and connected.
Each block of is a cycle, possibly degenerated to a single edge.
Every cutvertex of is connected to both and in .
The -tree of is a path.
Every non cutvertex of is connected to exactly one of and in , with the exception of exactly two vertices (one of them is ) which belong to the blocks of corresponding to degree- -nodes in the -tree of .
Since is formed by the vertices and edges of the outer face of , is outerplane. Also, since is triconnected, by Lemma 1, removing two vertices, namely and , cannot yield a disconnected graph. This proves Property . Property directly follows from the fact that is formed by the vertices and edges of the outer face of .
Concerning Property , we actually prove a stronger variant. Let be a cutvertex of , and consider a closed walk along the boundary of the outer face of . We claim that each occurrence of along this walk implies the existence, in , of one copy of edge or of one copy of edge . Since is simple, with this claim at hand, it follows that occurs exactly twice along the walk, and that is adjacent to both and . To prove the claim, let and be the vertex before and the vertex after an occurrence of along the walk, respectively. Namely, and belong to different biconnected components sharing only . We further distinguish two cases:
There is a face of that contains both and . Since is almost triangulated, such face is either a triangle or a marked quadrangle. In the former case, the edge would be part of , which is impossible, because is a cutvertex separating and . In the latter case, the marked face contains and a fourth vertex . By creftype 1, is neither nor , which implies that belongs to . This is again impossible, because is a cutvertex separating and .
There is no face of that contains both and . In this case, there exists an edge belonging to , such that does not belong to . Hence, is either or , as desired.
To show that the -tree of is a path (Property ), i.e., all the - and -nodes of the -tree have degree at most , we prove equivalently that each cutvertex of is shared by exactly two blocks and each block contains at most two cutvertices. The former follows from the fact that each cutvertex of occurs exactly twice in a closed walk along the boundary of the outer face of by Property . For the latter, we observe that if a block contains three cutvertices, , and , then the boundary of contains three vertex-disjoint paths that pairwise connect these cutvertices. Since and are connected to each other in and since by Property and are connected to each of , and , contains as a minor, which is impossible because is planar.
Concerning Property , we first show the existence of the two non-cut vertices of that are connected to both and . The first one is vertex by construction. For the other one consider the internal face of that is incident to the edge . By creftype 1, this face is not marked and thus the third vertex incident to this face, denoted by , is connected to both and . Since and as well as all the cutvertices of are connected to both and , by creftype 1 and the fact that the vertices of are on the outer face of , it follows that all other vertices of are connected to exactly one of or . Finally, if or belonged to a block whose -node in the -tree of has degree , then this block would contain three vertices all connected to both and , which would imply the existence of a minor in as in the proof of Property .
In view of Properties and of Lemma 2, we refer to as a chain of cycles and to its blocks as cycles, even when degenerated to single edges. Moreover, we denote by the non cutvertex of different from that is incident to both and , as specified in Property of Lemma 2.
Making each chain of cycles skinny. In order to incorporate the level- vertices that lie inside into the morph of , we perform a preliminary morph of to a strictly convex drawing of that is skinny, in the sense that it satisfies the following requirements with respect to the disk and the angle associated with the base edge derived from the morph of (see also Fig. 5 for an illustration).
Every cycle of is drawn inside the disk .
Every cycle of is drawn strictly convex.
The cutvertices of , as well as and , lie on the radius of .
For every cycle of and for every segment on its boundary, the smaller of the two angles formed at the intersection of the line through and the line through the segment is smaller than .
The existence of such a drawing is proven in the following lemma by means of a construction that exploits the properties of given in Lemma 2.
There exists a drawing of that is strictly convex, skinny, and topologically equivalent to .
We prove the statement by construction. Initially, we place and in the same positions as they are in . Further, we place the cutvertices of the chain of cycles as well as and on the radius in the order they appear in the chain to satisfy R.3. For each cycle of , we proceed as follows. Let and be the two vertices of that have already been placed on . Let and be two isosceles triangles sharing the same base , such that the third vertex of each of them lies inside and on opposite sides of and such that the internal angles at and at are smaller than ; refer to the colored triangles in Fig. 5. We place the vertices of that are incident only to (only to ) equidistant along a circular arc connecting and that lies completely inside (inside , respectively). By the definition of and , and also by the fact that the two circular arcs are drawn completely inside and , it follows that R.1, R.2, and R.4 are satisfied for the drawing of .
To complete the drawing of , we describe how to draw the subgraph of that is contained inside or on the boundary of such that every internal face of is strictly convex. Since is drawn convex in , it admits a strictly convex drawing for any given strictly convex drawing of its outer face . Thus, we can apply the algorithm in  to construct a strictly convex drawing of , whose outerface is the drawing of satisfying R.1-R.4. Finally, we add the edges incident to and that are contained inside to the resulting drawing, which does not introduce crossings due to R.4. This completes the construction of . Since every cycle in satisfies R.1–R.4 and since by creftype 1 all faces incident to and in are triangular, the drawing is strictly convex and skinny as desired. Since our construction and the algorithm in  maintain the cyclic order of the edges around each vertex, we have that is topologically equivalent to . This concludes the proof.
To describe the morph between and , we need some more work. Since both drawings are strictly convex and topologically equivalent, the preconditions of Theorem 3.2 are met. However, to ensure that this morph can be done independently for each piece of a kite, we further need to guarantee that vertices and do not move and that all vertices of remain inside throughout the morph. As stated in Theorem 3.2, this can be achieved if the (triangular) outer face is drawn the same in the two input drawings, which is not necessarily the case for and because of the position of (recall that and have the same position in and ). To this end, we augment and by adding a new vertex in the outer face of and connect it to , , and . Moreover, we place at the same position inside in both and so that the triangle formed by , , and contains all the other vertices of (in particular, ). The edge can always be drawn without crossings, as , , and were the vertices on the outer face of before. After this augmentation, we apply Theorem 3.2 to compute the desired morph of , and then we remove from the drawings.
Performing the global morph. Applying the above procedure for each piece of a kite yields a drawing of the supergraph of that is the union of and all the graphs corresponding to every piece of a kite . Observe that is composed of and the skinny drawing of every graph . To perform the global morph, recall that the vertices of the subgraph of follow the same trajectories as in the morph between and (which has been recursively computed). The level- vertices of each subgraph are moved inside , which again ensures that this can be done independently for each piece of a kite. In the following we describe the trajectories of one such subgraph. We denote this subgraph as and adopt the same notation as before.
Since the trajectories of and are specified by the morph between and , we only describe the trajectories of , i.e., the vertices of level ; see Fig. 6 for an example. The drawing of is a copy of rotated and translated so that the cutvertices of as well as and lie on the radius of perpendicular to , and the distance between and is the same as in . This ensures that the drawing of remains skinny, planar (by R.4), and strictly convex at every time instant.
Let be the drawing of obtained so far. The next step of the morph is to transform, for each subgraph , the current skinny drawing in to . By construction, and are topologically equivalent and strictly convex. Similarly as for , we insert vertex so that the outer face of is drawn the same in both and , which allows to apply Theorem 3.2 independently for each . The target drawing is obtained by removing the vertices and edges in and by reinserting the crossed edges in the marked faces. This concludes the proof of Theorem 1.1.
6 Implications of Theorem 1.1
In this section, we discuss the applicability of Theorem 1.1 by presenting meaningful families of -planar graphs that admit -kite-planar drawings.
An -vertex -planar graph has at most edges , and if it achieves exactly this density, then it is called optimal. Moreover, any -planar drawing of an optimal -planar graph is such that the uncrossed edges induce a plane triconnected quadrangulation , while each pair of crossing edges of is drawn inside a corresponding face of . When restricting to straight-line drawable -planar graphs, this bound is reduced to . Similarly to the general case, an optimal -planar straight-line drawing is one in which the uncrossed edges induce a plane triconnected graph whose every inner face is a quadrangle, while the outer face is a triangle . As a consequence, we obtain that each kite is empty and its kite edges are uncrossed. Therefore, any optimal -planar straight-line drawing is -kite-planar.
Another family of -planar graphs that recently attracted considerable attention are the IC-planar graphs [3, 15, 19, 29], which admit -planar drawings where the crossed edges induce a matching. Note that both the binding edges and the kite edges that are part of an IC-planar drawing are uncrossed. However, P.2 of Definition 1 may not be satisfied for those kite edges that are not part of the drawing. It follows that, if an IC-planar drawing is also kite-augmented , i.e., it contains all kite edges, then it is -kite-planar.
Overall, the following result is a corollary of Theorem 1.1.
There exists a morph between any pair of topologically-equivalent optimal -planar or kite-augmented IC-planar straight-line drawings.
We conclude this section with a remark. As already mentioned, Chambers et al.  studied morphs of toroidal graphs and asked to generalize their result to surfaces of higher genus. We note that, since an -vertex graph embeddable on a surface of genus has at most edges, while -vertex optimal -planar straight-line drawable graphs have edges, it follows that the latter do not admit an embedding (without edge crossings) on any surface of bounded genus. Thus, a solution to the open problem by Chambers et al. would not provide morphs of -kite-planar drawings.
7 Open Problems
We made a first step towards the problem of morphing pairs of non-planar drawings. Besides the general open problem of morphing any two such drawings , the main questions that stem from our research are as follows: (i) Is it possible to compute morphs of -kite-planar drawings where the vertex trajectories have bounded complexity? (ii) Regardless of the complexity, is it possible to extend Theorem 1.1 to all -planar drawings or to meaningful families of -planar drawings ()? (iii) Further families of beyond-planar graphs  could also be considered, for instance, does every pair of RAC drawings  admit a morph?
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