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On Point Set Embeddings for k-Planar Graphs with Few Bends per Edge

  • Michael KaufmannEmail author
Conference paper
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

We consider the point set embedding problem (PSE) for 1-, 2- and k-planar graphs where at most 1, 2, or k crossings resp. are allowed for each edge which greatly extends the well-researched class of planar graphs. For any set of n points and any given embedded graph that belongs to one of the above graph classes, we compute a 1-to-1 mapping of the vertices to the points such that the edges can be routed using only a limited number of bends according to the given embedding and the sequences of crossings. Surprisingly, for the class of 1-planar graphs the same results can be achieved as the best known results for planar graphs. Additionally for k-planar graphs, the bounds are also much better than expected from the first sight.

1 Introduction

Graph embeddings have been popular since quite some time in the area of combinatorics, graph algorithms and graph drawing. Variants include the bandwidth minimization problem [15] where the vertices should be ordered in unit distance on a line such that the total sum of the distances is minimized, but also graph embeddings, where a guest graph has to be embedded to a host graph under certain parameters like dilation, congestion, expansion, etc. [24].

In the geometric variant of the point set embedding (PSE) problem, a set S of points is assumed to be given, together with an input graph with certain properties. In the most simple version, S has exactly n points and the problem is to find a 1-to-1 mapping of the vertices to the points such that the corresponding straight-line edges are crossing-free. Techniques have been developed for the cases that the input graph is a tree [8, 20], an outerplanar graph [7, 29]. For the case of general planar graphs, the problem has been shown to be NP-hard. [9]. Another research direction concentrates on restricting the number of additional points that is needed to guarantee a planar straight-line embedding for any given planar graph. This has been called a ‘universal point set’. Unfortunately, the best upper bound for the general case is \(O(n^2)\) points as the universal point set [17], very far away from the lower bound of 1.098n [12]. Recently, smaller universal point sets of subquadratic size have been found for planar 3-trees [18], graphs of bounded pathwidth [4] and for the quite general class of so-called k-outerplanar graphs [3].

Another variant of the point set embedding problem with several applications assumes a given mapping of the vertices to the points. A linear number of bends per edge is sufficient and sometimes almost necessary [28] .

A well-recognized approach without a prescribed mapping but allowing bends on the edges has been developed by Kaufmann and Wiese [23], who developed an efficient PSE algorithm for any pointset of size n with only one bend per edge if the given plane graph is hamiltonian, and with only two bends per edge for general plane graphs. We will recall their algorithm in Sect. 2. We extend their technique and apply it to graph classes from a research direction which is called ‘beyond planarity’.

‘Beyond planarity’ is an informal term for a recent research direction in Graph Drawing, which is currently receiving a great deal of attention [19, 21, 25]. It mainly focuses on combinatorial and algorithmic aspects for classes of graphs that can be drawn on the plane while avoiding specific kinds of edge crossings; see, e.g., [14] for a survey.

Even quite long time ago, 1-planar graphs [30] have been considered as the family of graphs that admit drawings with at most one crossing per edge. Later generalizations include 2-planar graphs [27], 3-planar graphs [5, 26], k-planar graphs, where k crossings per edge are allowed. Others have more complicated restrictions like fan-planar graphs [22], where an edge is only allowed to be crossed by a set of adjacent edges from the same side (a fan), or fan-crossing-free graphs [10], where exactly this configuration was forbidden and only crossings by pairwise independent edges are allowed. Quasiplanar and k-quasiplanar graphs where no three resp. no k mutually crossing edges are allowed, received special considerations in the past due to their combinatorial structure [1, 2, 16] We finally mention the RAC graphs  [13], i.e. the family of graphs that admit polyline drawings, with few bends per edge, in which the angles formed at the edge crossings are \(90^\circ \).

Our Contribution: In this paper, we consider the PSE problem for 1-, 2- and k-plane graphs in the model, where the vertices-to-points mapping is not given and bends on the edges are allowed and should be minimized. In particular, after reviewing some theorems from the literature and the basic algorithm by Kaufmann and Wiese in Sect. 2, we show that an n-vertex 1-plane graph can be embedded on any given point set of size n with only 2 bends per edge in Sect. 3, matching the previously known best bound for plane graphs. In Sect. 4 we extend these techniques to 2-plane graphs where 2 crossings are allowed on each edge. Finally we demonstrate in Sect. 5 that even for general k-plane graphs very good results on the number of bends per edge can be achieved when the input graph is mapped on the given set of n points. We discuss open problems in Sect. 6.

2 Preliminaries

In this section, we introduce again the problem for our model. Then we recall some theorems and techniques from the literature, that we will use to obtain our results.

Let S be a set of n given disjoint points \(p_1 = (x_1,y_1), p_2=(x_2,y_2),..., p_n=(x_n,y_n)\) in the plane, and let \(G=(V,E)\) be an n vertex graph. The graph G is assumed to be simple and given as a topological drawing, i.e. a drawing of disjoint points as vertices and simple curves, representing the edges, such that there are no self-intersections, self-loops nor intersections of adjacent edges. We have to find a mapping of the vertices to the points and realizations of the edges as polygonal lines such that the topological drawing of the graph is preserved and the maximal number of bends on each edge is as small as possible.

For our approach, we will use the following three theorems, in particular Theorem 3.

Theorem 1

(Biedl, Kant, Kaufmann [6]). Any 2-connected planar graph without separating triangles can be triangulated efficiently without introducing new separating triangles. The new graph will be 4-connected.

Theorem 2

(Chiba, Nishizeki [11]). Any 4-connected planar graph has a hamiltonian cycle, which can be constructed in linear time.

Theorem 3

(Kaufmann, Wiese [23]). Any 4-connected plane graph can be embedded on any point set of the same size with only one bend per edge. For 3-connected plane graphs a corresponding result holds but the edges might have at most two bends.

We recall the algorithm of Kaufmann and Wiese [23]:

Let S be the given set of n points in the plane rotated such that the x-coordinates are disjoint and at least a certain constant space in x-direction exists between subsequent points. Let G a plane graph which can be assumed to be triangulated. If G is four-connected, we know that there exists a hamiltonian cycle \(C = v_1,...,v_n,v_1\), which can be efficiently computed [11]. We subsequently map the vertices \(v_i\) along the cycle to the points of S ordered from left to right. Consequently, the edges of the cycle C from \(v_1\) to \(v_n\) form an x-monotone polygonal line P. The closing edge \((v_n,v_1)\) is drawn with one bend above the polygonal line, such that the slopes of the ascending and the decreasing segment of the edge are drawn with the same slopes but with opposite sign. All the other edges can now be drawn easily either above the polygonal line inside of the cycle in a similar way as \((v_n,v_1)\) with one bend each, or below the line such that the bend is the lowest point of the edge which is drawn symmetrically as well.

More concretely, the slope is computed as follows: Define the parameter \(\rho = max_i \{\frac{|y_{i+1} - y_i|}{x_{i+1}-x_i}\}\) which denotes the steepest slope of a polygonal segment along the polygonal line P. To draw the edges not on the path with one bend symmetrically, we use slopes that are just slightly larger than \(\rho \) for ascending segments resp. slightly smaller than \(-\rho \) for descending segments. Simple adjustment of the slopes ensures the absence of crossings. Details can be found in [23]. This algorithm completes the first part.

The second part is more tricky. We assume that the graph is triangulated but not 4-connected, which implies that it has some separating triangles. Let \(T = \{u,v,w\}\) be one of the separating triangles with corresponding edges (uv), (vw), (wu). We break T by taking one of the edges, say (uv) and subdivide it by a dummy vertex d, a so-called b-dummy. Additionally, we triangulate the two quadrangular faces adjacent to d by two new so-called breaking edges incident to b. We apply this step to all the separating triangles and get a triangulated graph with some dummy vertices which has no separating triangles, i.e. which is 4-connected and hence has a hamiltonian cycle C. Clearly C also passes through the dummy vertex d. If it does not use both corresponding breaking edges, the dummy vertex is useless and can be removed without changing the remaining hamiltonian cycle C. Only if C passes through both breaking edges the dummy vertex is useful as the edge (uv) now consists of two subedges (ud) and (dv) which lie on different sides of the polygonal line P. If we apply now the basic algorithm of the four-connected case to the plane graph extended by the dummy vertices, we get at most 3 bends per edge, one for each part, and one at the dummy vertex. By changing the shape of the edges such that the edge segment incident to a dummy vertex consist of two vertical segments, the bend at the dummy vertex can be saved, such that each edge finally has at most 2 bends. Figure 1 shows the effect of the bend-saving technique. As it is explained in [23], the area consumption might be exponential in the size of the bounding rectangle of the point set S, see also Sect. 6.
Fig. 1.

The left figure shows the 3-bend version with symmetrical slopes. The bend at the dummy vertex can be saved by aligning the two middle segments vertically (right figure). Note that w might be to the left of the b-dummy, such that three bends seems unavoidable for the case of symmetrical slopes without the bend-saving technique.

In summary, edges that are incident to two original vertices have only one bend, and the edges that consist of two parts both incident to a dummy vertex have 3 bends, which can be reduced to two bends. In the following, we will heavily use this basic scheme, which will be extended appropriately.

We will planarize our non-planar input graph which will be given as a topological graph drawn in the plane by replacing the crossings by so-called c-dummy vertices. The corresponding edges will now consist of subedges between ‘real’ vertices and c-dummies. Those subedges will be called c-c edges if they are incident to two c-dummies. In case, the subedge is between a real vertex and a c-dummy, it is an r-c edge. Edges between a real and a b-dummy, are called r-b edges. If an edge is incident to two ‘real’ vertices, we call it an r-r edge.

3 1-Plane Graphs

In this section, we consider 1-plane graphs, i.e. graphs with at most one crossing per edge embedded in the plane. Let \(G = (V,E)\) be a 1-plane graph and S be a set of n points in the plane which is rotated such that the x-coordinates of the points are disjoint. As a first step, we replace the crossings of G by c-dummies and achieve a plane graph \(G'\). Now, we want to apply the algorithm of Kaufmann and Wiese to \(G'\). Because of 1-planarity, the edges in G might correspond to two r-c edges each in \(G'\). So, a first estimate indicates that the edges in G might have at most 5 bends, namely 2 bends for each subedge plus a bend at the corresponding c-dummy vertex. Unfortunately, the trick to reduce the bend at the dummy vertex cannot be applied directly.

Nevertheless we will prove the following theorem achieving a much better bound:

Theorem 4

Any 2-connected 1-plane graph can be embedded on any point set of the same size with at most two bends per edge.

Proof

After the replacement step of the crossings, we obtain the plane graph \(G'\). At each c-dummy, we have 4 incident r-c subedges. We triangulate \(G'\), e.g. by the algorithm of Biedl et al. [6] which has the property that new separating triangles are produced only if it is necessary. Next, we want to break the separating triangles. For each such triangle T, we select one of the edges to add a b-dummy vertex, which breaks the triangle T. Note that by 1-planarity, there is no edge between two c-dummies. So, if one of the edges is a r-c edge, then there are exactly two r-c edges in T. Clearly, the third edge is an r-r edge and we choose that edge to place the b-dummy vertex and which then is split into two r-b edges. Hence we never place an b-dummy on an r-c edges.

After the triangulation and breaking all the separation triangles by the b-dummies, we obtain \(G''\). \(G''\) finally is 4-connected such that a hamiltonian cycle C exists and we can apply approach of Kaufmann and Wiese. The hamiltonian cycle forms a monotone polygonal line P of straight-line segments between the points. Auxiliary points at appropriate positions, e.g. equidistantly spaced on the corresponding straight-line segment of the polygonal line P host the intermediate dummy vertices. Since only the edges of \(G'\) between two real vertices are split into at most two parts, we have on each original edge at most one dummy vertex, either a c-dummy or a b-dummy. Hence we have achieved already a bound of 3 for the number of bends per edge.

Next, we show how to save one of the three bends. If the dummy is a b-dummy, then the original technique of Kaufmann and Wiese can be applied and the middle bend can be saved. Note that this is quite straightforward also because b-dummies have degree 2. For the c-dummies, the four incident r-c edges make it more involved. In principle, we route the r-c edges such that the second segments (those that incident to the c-dummy) are nearly vertical, deviating from the vertical by a very small (\(\epsilon <0\)) angle to avoid unnecessary overlaps. Note that by chosing the bend points appropriately, this can be done without violating planarity. The preliminary crossing point is placed at the position of the c-dummy.

We distinguish 3 configurations, depending on how many of those four r-c edges are on the same sides of the polygonal line P defined by the hamiltonian path \((v_1,...,v_n)\). We have either 2, 3 or 4 of the r-c edges on different sides. The next figure shows some different configurations and our corresponding solutions.

  1. (a)

    Two of the r-c edges are on each side of P.

     

For this case, we see that the bend-saving technique of Kaufmann and Wiese can be applied directly. We leave the crossing at the c-dummy, and ensure that there are no bends there. Hence the claim to achieve 2 bends holds for this case.

  1. (b)

    Three of the r-c edges are on the same side of P.

     

This means that one of the two edges, call it e, consisting of two of the r-c edges does not cross P. Let \(b_1, b_2\) and \(b_3\) be the bends along e, where \(b_2\) is the c-dummy. Clearly, the triangle \(b_1, b_2, b_3\) is empty and is only crossed by one of the other two r-c edges incident to \(b_2\). Now, we omit \(b_2\) from e and add a straight-line segment from \(b_1\) to \(b_3\). Hence we save the middle bend. The crossing point is now at another position, but since its position is not prescribed, this is not problematic. The other edge which crosses P at the c-dummy, can be routed with only 2 bends as in Case (a). Note that the topology of the crossing is preserved.

  1. (c)

    All four r-c edges are on the same side of the polygonal line.

     

We apply the same bend-saving trick as in the case (b) but now for both edges. The properties that the corresponding triangles described by the 3 bends are empty hold as well. Hence we can save the two middle bends achieving the two-bends bound. See also the right-hand side of Fig. 2. Clearly, the crossing still exists since we do not change the beginning and end of the two edges.    \(\square \)

Fig. 2.

Two different configurations (a) and (c) for a crossing c with the corresponding incident edge segments. On the right-hand side, bend-saving is applied, and the dashed segments denote the new middle segments

In the next section, we examine how much of the techniques can be transfered to the 2-planarity case, where there might be two crossings on each edge.

4 2-Plane Graphs

Let \(G = (V,E)\) be a 2-connected 2-plane graph and S be a set of n points in the plane which is rotated to provide disjoint x-coordinates of the points. As before, we replace the crossings of G by so-called c-dummies and achieve a plane graph \(G'\). Before executing the triangulation step, we analyse the existing separating triangles and restrict the set of edges that will be split by b-dummy vertices.

Lemma 1

In a separating triangle in \(G'\), there is an r-r edge.

Proof

We first assume that there is a separating triangle \(T = \{u,v,w\}\) in \(G'\) such that it contains only c-c edges, hence uv and w are c-dummies. Hence the component inside of the separating triangle has either no or only one edge connecting it to the three crossings, which is a contradiction to the 2-planarity assumption, as the edge might have 3 crossings, or there are two edges from the inside of T to the crossings, then T forms a selfloop of an edge consisting of at least three parts.

Second, we assume that there is a separating triangle T, that has an r-c edge, but no r-r edge. Note that then it has exactly two incident r-c edges, and one c-c edge. Let v be the real vertex in T. By 2-connectivity, the component has at least two different connections to T. If both c-dummies are adjacent to the inside component, then T is a selfloop edge consisting of three subedges and incident to vertex v. If only one c-dummy, say w, is adjacent to the inside component, then the subedges (vw) and (wu) belong to the same edge, and this edge crosses the other edge that starts at the real vertex with the subedge (vu), which is then a contradiction to the simplicity of the drawing, as two incident edges intersect. Figure 3 gives an overview of the different cases.    \(\square \)

Fig. 3.

Different configurations for a separating triangle with real vertex v and c-dummies u and w. The circle inside the triangle denotes an inside component.

Now we extend \(G'\) by new triangulation edges as planned before [6]. There might occur some separating triangles that existed already in \(G'\), or some new separating triangles that include some new (fictitious) triangulation edges. To break the later separating triangles, we use the fictitious edges to place the b-dummies onto, since those edges will not be drawn anyway and hence we do not care about the number of bends they will have.

We only focus on the ‘original’ separating triangles in \(G'\) and we apply Lemma 1. Hence we split these separating triangles only by b-dummy vertices on r-r edges. We again have a 4-connected graph \(G''\) with a hamiltonian cycle C which then implies the order of the real vertices in which those vertices will be mapped along the polygonal monotone path P. The dummy vertices are placed in between respectively. Since b-dummies are only placed at r-r edges and fictitious edges, the bend-saving technique of Kaufmann and Wiese can be applied to make sure that the number of bends on the real edges is at most two. The most critical edges are those edges with two crossings, as they are subdivided by two c-dummies. They consist of three subedges with two segments respectively with slopes slightly larger than parameter \(\rho \) for ascending segments, and slightly smaller than \(-\rho \) for descending segments. Without the bend-saving technique, the intermediate c-dummies would imply two more bends, and hence 5 bends in total.

Applying the bend-saving technique realizes each subedge by a nearly-vertical segment and the second segment with a large (positive or negative) slope. Unfortunately, not both of the bends at the c-dummies can be saved by the technique if the middle subedge lies on the other side of the polygonal path P, i.e. if we have to cross P twice at the c-dummies. Corresponding bends can only be saved if we have (almost)-vertical segments at the c-dummies. This can only be realized at one of them. Hence the number of bends on each edge can be upperbounded by 4.

Theorem 5

Any 2-connected 2-plane graph can be embedded on any point set of the same size with at most four bends per edge.

In Fig. 4 we give an example of an edge with two crossings c1 and c2, where the hamiltonian cycle is crossed twice and the applicability of the bend-saving technique is limited. This effect generalizes also to the case of higher number of crossings per edge (see Sect. 5) if the edge crosses the hamiltonian cycle upto k times. In the next section, we will transfer the insights about the problems in 2-planar graphs to general k-planar graphs for \(k \ge 2\).
Fig. 4.

A sketch of the routing of an edge (vw) which is crossed twice, and the vertices and c-dummies come in a certain order along the hamiltonian cycle. The bend at c1 can be saved, but the one at c2 cannot. This indicates that saving the bends can eventually be applied only at every other c-dummy.

5 k-Plane Graphs

Let G be a k-plane graph for \(k \ge 2\) given as a topological drawing with at most k crossings on each edge and let S be a set of n points in the plane which is rotated to provide disjoint x-coordinates of the points. Again, the crossings of G are replaced by the c-dummies such that a plane graph \(G'\) has been constructed. Note that each edge which has l crossings is subdivided into two r-c edges and \(l-1\) c-c edges. Before executing the triangulation step, we will try to make similar observations as in the 2-planarity case such that we restrict the set of edges that will be split by b-dummy vertices to remove the separating triangles.

Lemma 2

In a separating triangle in \(G'\), there is an r-r edge.

Proof

We assume that there is a separating triangle \(T = \{u,v,w\}\) in \(G'\) without any r-r edge.

By 2-connectivity, at least two of the (dummy) vertices have one or more connections to the inside of T. As there are no r-r edges in T, there are at least two c-dummies in T. Only one of three cases might occur: Either the three subedges of T form a loop in G, or there are two edges in G that cross each other twice in G, or one of the vertices of T is a real vertex v, and there are two edges incident to v that cross each other. All three cases lead to a contradiction to simplicity assumptions for the topological embedding of G. Some of the cases can be checked in Fig. 3.    \(\square \)

Next we apply the triangulation procedure adding some (fictitious) edges. Then we extend the graph by breaking separating triangles either placing b-dummy vertices on fictitious edges (if they exist), or we can use the r-r edges within the ‘original’ separating triangles to destroy those separating triangles by placing b-dummies. Then we proceed with the algorithm of Kaufmann and Wiese for 4-connected plane graphs, guided by the hamiltonian cycle C which we determine by the algorithm of Chiba/Nishizeki [11], and parameter \(\rho \). Without the bend-saving tricks, we achieve for an edge with k crossings, hence with k c-dummies and \(k+1\) subedges at most \(2k+1\) bends.

Theorem 6

Any 2-connected k-plane graph can be embedded on any point set of the same size with at most \(2k+1\) bends per edge.

Further improvements can be achieved applying the bend-saving techniques that we developed before. Unfortunately, we were not able to save all the bends at the c-vertices. It can successfully be done when two of the subedges consequently are on the same side of the polygonal line formed by the monotone path P along the hamiltonian cycle. The worst case appears if subsequent subedges alternate between the sides of P. In that case, we can save only every second bend at a c-vertex, as we have to use two nearly-vertical segments at this vertex. When we have fulfilled this condition for a c-vertex, we cannot fulfill it for the next c-vertex, or we have to use an additional bend, cf. Fig. 4 where the case for 2 crossings is sketched. Hence we can conclude

Theorem 7

Any 2-connected k-plane graph can be embedded on any point set of the same size with at most \(k+1 + \lfloor k/2 \rfloor \le \frac{3}{2} k +1\) bends per edge.

6 Discussions and Conclusions

We have shown how to effectively apply the algorithm of Kaufmann and Wiese [23] to 1-planar, 2-planar and finally k-planar embedded graphs for \(k \ge 2\). Note that the area consumption is exponential even in the basic algorithm for plane graphs if the bend-saving techniques are applied and the bound of only two bends should be maintained. For the case of three bends where the parameter \(\rho \) is used to control the size of the slopes, and using the assumption of an integer grid, the authors mention an upper bound of \(O(W^3)\) for the area, where W denotes the width of the bounding box of the points. We give the corresponding bounds for the graph classes here:

Theorem 8

Any 2-connected k-plane graph can be embedded on any point set of the same size with at most \(2k+1\) bends per edge using only a drawing area of polynomial size. In particular, for \(k=1\), we have 3 bends per edge, and for \(k=2\), we have 5 bends per edge.

We conclude with some open problems:
  1. 1.

    Improve the bounds for the number of bends per edge for 2-plane graphs from four to three. This then might imply an improvement of the bounds for general k-plane graphs as well.

     
  2. 2.

    The area issue should be discussed in more detail, even for the crossing-free case. Is it possible to keep the area of polynomial size and still have only two bends per edge. Develop lower bounds.

     
  3. 3.

    Consider the special case of outer-1-plane graphs. Note that there are efficient algorithms for straight-line embeddings of outerplanar graphs on pointsets in general positions. Can this result be transfered to outer-1-plane graphs? What are the corresponding bounds for outer-2-plane and outer-k-planar graphs?

     

Notes

Acknowledgement

The author wishes to thanks the participants of the GNV workshop in Heiligkreuztal 2018 for inspiring discussions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Wilhelm-Schickhard-Institut für InformatikUniversität TübingenTübingenGermany

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