Beyond-Planarity: Density Results for Bipartite Graphs

12/28/2017 ∙ by Patrizio Angelini, et al. ∙ KIT 0

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani 1966) for extremal problems on geometric graphs, but is also related to graph drawing problems where visual clutter by edge crossings should be minimized (e.g., Huang et al. 2008) that could negatively affect the readability of the drawing. Different types of forbidden crossing configurations give rise to different families of nearly-planar graphs. Most of the literature focuses on Turán-type problems, which ask for the maximum number of edges a nearly-planar graph can have. Here, we study this problem for bipartite topological graphs, considering several types of nearly-planar graphs, i.e., 1-planar, 2-planar, fan-planar, and RAC graphs. We prove bounds on the number of edges that are tight up to small additive constants; some of them are surprising and not along the lines of the known results for non-bipartite graphs. Our findings lead to an improvement of the leading constant of the well-known Crossing Lemma for bipartite graphs, as well as to a number of interesting research questions on topological graphs.

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1 Introduction

Planarity has been one of the central concepts in the areas of graph algorithms, computational geometry, and graph theory since the beginning of the previous century. While planar graphs were originally defined in terms of their geometric representation (i.e., a topological graph is planar if it contains no edge crossing), they exhibit a number of combinatorial properties that only depend on their abstract representation. To cite only some of the most important landmarks, we refer to the characterization of planar graphs in terms of forbidden minors, due to Kuratowski [51], to the existence of linear-time algorithms to test graph planarity [21, 26, 44], to the Four-Color Theorem [10, 11], and to the Euler’s polyhedron formula, which can be used to show that -vertex planar graphs have at most edges.

For the applicative purpose of visualizing real-world networks, however, the concept of planarity turns out to be restrictive. In fact, graphs representing such networks are generally too dense to be planar, even though one can often confine non-planarity in some local structures. Recent cognitive experiments [45] show that this does not affect too much the readability of the drawing, if these local structures satisfy specific properties. In other words, these experiments indicate that even non-planar drawings may be effective for human understanding, as long as the crossing configurations satisfy certain properties. Different requirements on the crossing configurations naturally give rise to different classes of topological or geometric, i.e. straight-line, nearly-planar graphs. Beyond-planarity is then defined as a generalization of planarity, which encompasses all these graph classes. Early works date back to the 1960’s [14] in the field of extremal graph theory, and continued over the years [5, 8, 19, 50, 55]; also due to the aforementioned experiments, a strong attention on the topic was recently raised [53, 43, 47], which led to many results described below.

Some of the most studied nearly-planar graphs include: (i) k-planar graphs, in which each edge is crossed at most times [2, 19, 49, 54, 55, 56], see Fig. 0(a); (ii) k-quasiplanar graphs, which disallow sets of pairwise crossing edges [4, 5, 38], see Fig. 0(b); (iii) fan-planargraphs, in which no edge is crossed by two independent edges or by two adjacent edges from different directions  [15, 17, 18, 48], see Fig. 0(c); (iv) RAC graphs, in which edge crossings only happen at right angles [33, 34, 35], see Fig. 0(d). Two notable sub-families of -planar graphs are the IC-planar graphs [23, 60], in which crossings are independent (i.e., no two crossed edges share an endpoint), and the NIC-planar graphs [59], in which crossings are nearly independent (i.e., no two pairs of crossed edges share two endpoints). Other families include fan-crossing free graphs [24], planarly-connected graphs [3], and bar-k-visibility graphs [27, 37].

(a) 1-planar
(b) 3-quasiplanar
(c) fan-planar
(d) RAC
Figure 1: Different forbidden crossing configurations.

From the combinatorial point of view, the main question concerns the maximum number of edges for a graph in a certain class. This extreme graph theory question is usually referred to as a Turán-type problem [20]. Tight density bounds are known for several classes [33, 48, 54, 55, 59, 60]; a main open question is to determine the density of -quasiplanar graphs, which is conjectured to be linear in for any fixed  [1, 5, 8, 38]. Works on finding tight bounds on the edge density of -,-, - and -planar graphs have led to corresponding improvements on the leading constant of the well known Crossing Lemma [6, 52]; refer to [2, 54, 55]. Another combinatorial question is to discover inclusion relationships between classes [9, 18, 23, 28, 35, 37, 40].

From the complexity side, in contrast to efficient planarity testing algorithms [44], recognizing a nearly-planar graph has often been proven to be NP-hard [12, 15, 18, 23, 39]. Polynomial-time testing algorithms can be found when posing additional restrictions on the produced drawings, namely, that the vertices are required to lie either on two parallel lines (2-layer setting; see, e.g., [17, 18, 32, 30]) or on the outer face of the drawing (outer setting; see, e.g., [13, 15, 29, 41, 42]).

Each of these variants define new graph classes, which have also been studied in terms of their maximum density, e.g. [15, 17, 18]. Another natural restriction, which has been rarely explored in the literature, is to pose additional structural constraints on the graphs themselves, rather than on their drawings. For 3-connected 1-plane graphs, Alam et al. [7] presented a polynomial-time algorithm to construct 1-planar straight-line drawings. Further, Brandenburg [22] gave an efficient algorithm to recognize optimal 1-planar graphs, i.e., those with the maximum number of edges.

For the important class of bipartite graphs, very few results have been discovered yet. From the density point of view, the only result we are aware of is a tight bound of edges for bipartite -planar graphs [25, 46]. Didimo et al. [32] characterize the complete bipartite graphs that admit RAC drawings, but their result does not extend to non-complete graphs.

General Bipartite
Graph class Bound (tight)  Ref.  Lower bound  Ref.  Upper bound  Ref.
IC-planar: [60] Thm.1 Thm.2
NIC-planar: [59] Thm.3 Thm.4
-planar: [56] [25] [25]
RAC: [33] Thm.5 Thm.6
-planar: [55] Thm.9 Thm.10
fan-planar: [48] Thm.7 Thm.8
-planar: [16] Sec.10
-planar: [2] Thm. 12
Table 1: Summary of our results (from sparse to dense); the bound with an asterisk () is not tight.

1.1 Our contribution

Along this direction, we study in this paper several classes of nearly-planar bipartite topological graphs, focusing on Turán-type problems. Table 1 shows our findings. Note that the new bound on the edge density of bipartite -planar graphs leads to an improvement of the leading constant of the Crossing Lemma for bipartite graphs from  [2] to (see Theorem 11), as well as to a new bound for the edge density of bipartite -planar graphs (see Theorem 12). Additionally, our results unveil an interesting, and somehow unexpected, tendency in the density of -planar bipartite topological graphs with respect to the one of general -planar graphs. At first sight, the differences seem to be around , as it is in the planar and in the 1-planar cases (i.e., ). This turns out to be true also for RAC and fan-planar graphs. However, for the cases of IC- and NIC-planar graphs, and in particular for 2-planar graphs, the differences are surprisingly large. Considering ratios between the bounds instead of the differences, the results are even more unexpected.

Another notable observation from our results is that, in the bipartite setting, fan-planar graphs can be denser than -planar graphs, while in the non-bipartite case these two classes have the same maximum density, even though none of them is contained in the other [17]. In Section 10 we discuss a number of open problems that are raised by our work.

2 Methodology

We focus on five classes of bipartite nearly-planar graphs: IC-planar, NIC-planar, RAC, fan-planar and -planar graphs; refer to Sections 48

. To estimate the maximum edge density of each class we employ different counting techniques.

  1. For the class of bipartite IC-planar graphs, we apply a direct counting argument based on the number of crossings that are possible due to the restrictions posed by IC-planarity.

  2. Our approach is different for the class of bipartite NIC-planar graphs. We show that a bipartite NIC-planar graph of maximum density contains a set of uncrossed edges forming a plane subgraph whose faces have length . The density bound is obtained by observing that one can embed exactly one crossing pair of edges inside each facial -cycle.

  3. To estimate the maximum number of edges of a bipartite RAC graph, we adjust a technique by Didimo et al. [33], who proved the corresponding bound for general RAC graphs.

  4. For bipartite fan-planar graphs, our technique is more involved. We first examine structural properties when these graphs are maximal. Then, we show how to augment any of these graphs (by appropriately adding vertices and edges) such that it contains as a subgraph a planar quadrangulation. Based on this property, we develop a charging scheme which charges edges involved in fan crossings to the corresponding vertices, so that the difference between the degree of a vertex and its charge is at least 2. This implies that there are at least as many edges in the quadrangulation as in the rest of the graph.

  5. Our approach for bipartite -planar graphs follows similar lines. We show that maximal bipartite 2-planar graphs have a planar quadrangulation as a subgraph. We then use a counting scheme based on an auxiliary directed plane graph, defined by appropriately orienting the dual of the quadrangulation, which describes dependencies of adjacent quadrangular faces posed by the edges that do not belong to the quadrangulation.

3 Preliminaries

We consider connected topological graphs, i.e., drawn in the plane with vertices represented by distinct points in and edges by Jordan curves connecting their endvertices, so that: (i) no edge passes through a vertex different from its endpoints, (ii) no two adjacent edges cross, (iii) no edge crosses itself, (iv) no two edges meet tangentially, and (v) no two edges cross more than once. A graph has no self-loops or multiedges. Otherwise, it is a multigraph, for which we assume that the bounded and unbounded regions defined by self-loops or multiedges contain at least one vertex in their interiors, i.e., there are no homotopic edges.

We refer to a nearly-planar graph with vertices and maximum possible number of edges as optimal. Also, we denote by a maximal plane subgraph of on the same vertex-set as , i.e., with the largest number of edges such that in the drawing of  inherited from  there exists no two edges crossing each other. We call the planar structure of . Let be a face of . We say that is simple if for each , and it is connected if edge exists for each (indices modulo ).

4 Bipartite IC-planarity

In this section, we give a tight bound on the density of bipartite IC-planar graphs.

Theorem 1.

There exist infinitely many bipartite -vertex IC-planar graphs with edges.

Proof.

Fig.1(a) shows a construction that yields -vertex bipartite IC-planar graphs with edges. The graph is composed of a quadrangular grid of size , which we wrap around a cylinder by identifying the vertices of its topmost row with the ones of its bottommost row. Thus, the two bases of the cylinder are faces of length

. Now, observe that each vertex in our construction participates in exactly one crossing. Hence, the number of skewed edges of our construction is exactly

. Since a planar graph with vertices, whose faces are of length , has exactly edges, it follows that the constructed graph has exactly  edges. ∎

Theorem 2.

A bipartite -vertex IC-planar graph has at most edges

Proof.

Our proof is an adjustment of the one for general IC-planar graphs [60]. Let  be a bipartite -vertex optimal IC-planar graph. Let be the number of crossings of . Since every vertex of  is incident to at most one crossing, . By removing one edge from every pair of crossing edges of , we obtain a plane bipartite graph, which has at most edges. Hence, the number of edges of  is at most . ∎

5 Bipartite NIC-planarity

We continue our study with the class of bipartite NIC-planar graphs.

Theorem 3.

There exist infinitely many bipartite -vertex NIC-planar graphs with edges.

Proof.

Fig.1(b) shows a construction that yields -vertex bipartite NIC-planar graphs with edges. Again, the graph is composed of a quadrangular grid of size , which we wrap around a cylinder by identifying the vertices of its topmost row with the ones of its bottommost row. Thus, the two bases of the cylinder are faces of length . Now, observe that each vertex in our construction participates in exactly two crossings, except for exactly four vertices (marked by dotted circles) participating only in one crossing. Hence, the number of skewed edges of our construction is exactly . Since a planar graph with vertices, whose faces are of length , has exactly edges, it follows that the constructed graph has exactly  edges. ∎

(a) IC-planar
(b) NIC-planar
Figure 2: Bipartite -vertex IC- and NIC-planar graphs with (a)  and (b)  edges.
Theorem 4.

A bipartite -vertex NIC-planar graph has at most edges.

Proof.

Let be a bipartite optimal NIC-planar graph with vertices; among such graphs, we assume that is one with the maximum number of edges that are not involved in any crossing. Namely, is such that the plane (bipartite) subgraph obtained by removing every edge that is involved in a crossing in has maximum density.

Next, we claim that each face of containing two crossing edges in is connected and has length (hence, every face of has length either , if it contains two edges crossing in , or otherwise due to bipartiteness and maximality). Consider any pair of edges and that cross in ; let and belong to the same partition of , which implies that and belong to the other partition. By -planarity and by the optimality of , we can assume that edges and belong to , and in particular that there exist copies of these edges in such that the two regions delimited by , , and , and by , , and , respectively, do not contain any vertex in their interior. We now show that there exists a vertex and two edges and of such that the region delimited by , , , and does not contain any vertex in its interior. Consider the edge such that edges , , and appear consecutively around . If belongs to , then we can assume that also belongs to , due to the maximality of , hence satisfying the required property. Otherwise, suppose for a contradiction that is crossed by some other edge in ; observe that is not an endpoint of , due to NIC-planarity. We then remove , hence making belong to , and we add a copy of edge to so to satisfy the required property. Namely, we draw this edge as a curve that starts at , follows , then , and finally , till reaching . Note that, if there exists another copy of edge in , it did not cross edge before its removal; hence, this copy of is not homotopic with the new copy we added. Since this operation results in a graph with the same number of edges of , and in a graph with more edges than , we have a contradiction to the maximality of . Applying the same arguments we can prove that there exists a vertex such that edges and belong to and the region delimited by , , , and does not contain any vertex in its interior. This concludes the proof of our claim.

Let and be the number of vertices and edges of , respectively. Clearly, . Let also and be the number of faces of length and in , respectively. We have that . By Euler’s formula, we also have that . Combining these two equations, we obtain: . So, in total the number of edges of is . By Euler’s formula, the number of faces of length of a planar graph is at most , which implies that has at most edges. ∎

6 Bipartite RAC Graphs

In this section, we continue our study on bipartite beyond-planarity with the class of RAC graphs. We prove an upper bound on their density that is optimal up to a constant of .

Theorem 5.

There exist infinitely many bipartite -vertex RAC graphs with edges.

Proof.

For any , we recursively define a graph by attaching six vertices and edges to as in Fig. 2(a); the base graph is a hexagon containing two crossing edges (see Fig. 2(c)). So, has vertices and edges. Fig. 2(b) shows that is RAC: if has been drawn so that its outerface is a parallelogram (gray in Fig. 2(b)), then we can augment it to a RAC drawing of in which the outerface is a parallelogram whose sides are parallel to the ones of . The bound is obtained by adding one more edge in the outerface of by slightly “adjusting” its drawing; see Fig. 2(d). ∎

(a)
(b)
(c)
(d)
Figure 3: Construction for a bipartite -vertex RAC graph with edges.
Theorem 6.

A bipartite -vertex RAC graph has at most edges.

Proof.

We use an argument similar to the one by Didimo et al. [33] to prove the upper bound of edges for general RAC graphs: Let be a (possibly non-bipartite) RAC graph with vertices. Since does not contain three mutually crossing edges, as in [33] we can color its edges with three colors (r, b, g) so that the crossing-free edges are the r-edges, while b-edges cross only g-edges, and vice-versa. Thus, the subgraphs , consisting of only r- and b-edges, and , consisting of only r- and g-edges, are both planar. Didimo et al. [33, Lemma 4] showed that each face of has at least two r-edges, by observing that if this property did not hold, then the drawing could be augmented by adding r-edges. Thus, the number of b-edges is at most , where is the number of edges in the outer face of . Suppose now that  is additionally bipartite. We still have , but in this case holds (by bipartiteness). Hence, . Since is bipartite and planar, it has at most edges (i.e., ). By combining the latter two inequalities, we obtain that the total number of edges of is at most , as desired. ∎

7 Bipartite Fan-Planarity

We continue our study with the class of fan-planar graphs. We begin as usual with the lower bound (see Theorem 7), which we suspect to be best-possible both for graphs and multigraphs. For fan-planar bipartite graphs, we prove an almost tight upper bound (see Theorem 8).

Theorem 7.

There exist infinitely many bipartite fan-planar (i) graphs with vertices and exactly edges, and (ii) multigraphs with vertices and exactly edges.

Proof.

Recall that a fan-planar multigraph is a graph, possibly with multiedges, that admits a drawing with no homotopic edges in which for every edge all edges crossing have a common endpoint, which is moreover on the same side of (as we consider bipartite fan-planar graphs here, no loops occur). For the first part it suffices to consider , , which has edges and is known to be fan-planar for any  [48]. Another exceptional example is , that is, minus an edge, which has vertices and edges; see Fig. 3(c).

(a)
(b)
(c)
Figure 4: Fan-planar drawings of bipartite multigraphs with vertices and edges: (a)  with four additional multiedges (thick), (b)  with additional multiedges (thick), (c)  with four additional multiedges (thick).

For the second part we observe that one can add four additional multiedges to the fan-planar drawings of and as illustrated in Figs. 3(a) and 3(c). Another class of examples is given by , , to whose planar drawing one can add additional multiedges in a fan-planar way as illustrated in Fig. 3(b), giving a fan-planar multigraph with vertices and edges. ∎

Remark.

Our upper bound (see Theorem 8) implies that the complete bipartite graphs and are not fan-planar. This is a big improvement over previous results, as the upper bound for general fan-planar graphs only implies that is not fan-planar, while it gives no guarantee for any . However, we suspect that already is not fan-planar, which would follow from an upper bound of edges for -vertex fan-planar graphs and would resolve all remaining cases of complete bipartite graphs.

To prove the upper bound, consider a bipartite fan-planar graph with a fixed fan-planar drawing. W.l.o.g. assume that is edge-maximal and connected, and , are the two bipartitions of . Throughout this section we shall denote vertices in by , , or for some index , and similarly vertices in by , , or . By fan-planarity, for each edge of all edges crossing have a common endpoint (which also lies on the same side of ). We call an -edge (respectively, -edge) if this vertex lies in (respectively, ).

A cell of some subgraph of is a connected component of the plane after removing all vertices and edges in ; see also [48]. The size of , denoted by is the total number of vertices and edge segments on the boundary of , counted with multiplicities.

Lemma 1 (Kaufmann and Ueckerdt [48]).

Each fan-planar graph admits a fan-planar drawing such that if is a cell of any subgraph of , and , then contains no vertex of in its interior.

We choose a fan-planar drawing of with the property given in Lemma 1.

Corollary 1.

If an edge , with and , is crossed in point by an -edge , then every edge crossing between and is an -edge that is moreover crossed by each edge that crosses .

Proof.

Let be the common endpoint of all edges crossing and be the -edge crossing in . Let be an edge that crosses between and . If is not an -edge, it is crossed by an edge with . The -edge is not crossed by ; see Fig. 4(a). But then there is a cell bounded by vertex and segments of , and , which contains vertex in its interior, contradicting Lemma 1. Symmetrically, if there is an edge that crosses but not (see Fig. 4(b)), then there is a cell bounded by vertex and segments of , and , which contains vertex , again contradicting Lemma 1. ∎

(a)
(b)
(c)
Figure 5: Illustration of (a)-(b) the proof of Corollary 1, and (c) Lemma 3.

Kaufmann and Ueckerdt [48] derive Lemma 1 from the following lemma.

Lemma 2 (Kaufmann and Ueckerdt [48]).

Let be given with a fan-planar drawing. If two edges and cross in a point , no edge at crosses between and , and no edge at crosses between and , then and are contained in the same cell of .

By the maximality of we have in this case that is an edge of , provided and lie in distinct bipartition classes. We can use this fact to derive the following lemma.

Lemma 3.

Let and be two crossing edges. If and are both -edges or both -edges, then is also contained in and can be drawn so that it crosses only edges that also cross . If is an -edge and is a -edge, then is also contained in and can be drawn crossing-free.

Proof.

First assume that and are both -edges; the case where and are both -edges is analogous. Let be the crossing point on that is closest to . Since is an -edge crossing , the edge crossing at (possibly ) is incident to . Now either or the subgraph of consisting of , and (and their vertices) has one bounded cell of size , which by Lemma 1 contains no vertex of . In both cases it follows that every edge of crossing between and , also crosses , and hence ends at (since is an -edge crossing ). We can conclude that drawing an edge from along to and then along to does not violate fan-planarity; see Fig. 4(c) for an illustration. Thus, by the maximality of , edge is contained in .

Now assume that is an -edge and is a -edge. Let be the crossing point of and . By Lemma 2, and lie on the same cell in and hence, by the maximality of , we have that the edge is contained in and can be drawn crossing-free. ∎

We are now ready to prove the main theorem of this section.

Theorem 8.

Any -vertex bipartite fan-planar graph has at most edges.

Proof.

We start by considering the planar structure of , i.e., an inclusion-maximal subgraph of whose drawing inherited from is crossing-free. Let and be the set of all -edges and -edges, respectively, in . Each is crossed by a non-empty (by maximality of ) set of edges in with common endpoint , and we say that charges . Similarly, every charges a unique vertex .

For any vertex in , let denote the number of edges in charging . Moreover, for a multigraph containing , let denote the degree of in , i.e., the number of edges of incident to . Our goal is to show that for every vertex of we have . However, this is not necessarily true when is not connected or has faces of length  or more. To overcome this issue, we shall add in a step-by-step procedure vertices and edges (possibly parallel but non-homotopic to existing edges in ) to the plane drawing of such that:

  1. the obtained multigraph is a planar quadrangulation,

  2. the drawing of the multigraph is again fan-planar, and

  3. each new vertex is added with three edges to other (possibly earlier added) vertices.

To find , we first prove in the following claim that if is not a quadrangulation, we can add either one new edge or one new vertex with three new incident edges that do not cross any edge of . Moreover, the resulting multigraph (which may have parallel but non-homotopic edges) will still be bipartite and its drawing will still be fan-planar.

Claim 1.

If is not a quadrangulation, one can add either one new edge or one new vertex with three new incident edges to the drawing of , such that the resulting multigraph is still bipartite, the resulting drawing is still fan-planar, and the new edges do not cross any edge of .

Proof.

First assume that is not connected. Then, there exists an edge in where and lie in different connected components of (w.l.o.g. ). As , there is an edge in crossing . By symmetry, we may assume that and lie in different components of . Furthermore, w.l.o.g.  is the edge of whose crossing with is closest to . We distinguish four cases.

Case 1. and are -edges.

Then by Lemma 3 there is an edge in and it can be drawn so that it crosses only edges that cross between and ; see Fig.5(a). None of the latter edges are in by the choice of . Hence can be added to , contradicting the maximality of and that and are in different components of .

Case 2. and are -edges.

Again by Lemma 3 there is an edge in and this time it can be drawn so that it crosses only edges that cross between and ; see Fig.5(b). None of the latter edges are in as they cross which is in . Hence can like in Case 1 be added to , arriving at the same contradiction.

(a)
(b)
(c)
(d)
Figure 6: Illustrations for the proof of Theorem 8. Edges in are drawn thick, newly added vertices and edges are drawn in red. In (fig:-fan-planar-connectivity-4) an edge of the form , , is added only if the edge is not in .
Case 3. is a -edge and is an -edge.

Here Lemma 3 immediately gives that there is an edge in that can be drawn without crossings. Hence, as in the cases before, is in , putting and in the same component of .

Case 4. is an -edge and is a -edge.

This case is more elaborate. Consider a point in the plane very close to and on the -side of and the -side of ; see Fig.5(c).

First, we claim that every edge crossing between and is an -edge. In fact, as (by choice of ), it is crossed by some edge in , but if were a -edge, then by Corollary 1 this edge in would also cross , which is impossible. Hence we can draw a curve from to crossing only -edges that also cross . Thus, does not cross any edge of .

Second, we claim that every edge crossing between and is an -edge. In fact, this follows from Corollary 1 and the fact that is an -edge. Hence we can draw a curve from to crossing only -edges that also cross . Note that does not cross any edge of .

Now consider the cell of containing point . If the boundary of contains some vertex from , we can extend and to two edges and respectively without creating any further crossings. Note that this drawing is again fan-planar. These two edges do not cross any edge of , and as and are in different components of , at least one such edge is not already present in and we are done.

If the boundary of contains some vertex from , we can add a new vertex to at point and draw edges , , and ; see Fig.5(c). The resulting drawing is still fan-planar and new edges do not cross any edge of , as desired.

Finally, we assume that contains, expect for possibly , no vertex of . Let us start tracing beginning with and following towards . At some point we encounter a crossing of with another edge . Since is a -edge, for some . We follow along towards and encounter a crossing of with another edge ; see Fig.5(d). If would be incident to , then by Corollary 1 would cross between and and hence would be a -edge. Then, again by Corollary 1, every edge crossing would also cross , which is in , which gives that , contradicting the choice of . Thus, is incident not to but to , making a -edge. Let denote the other endpoint of , , possibly . If is a -edge, then every edge crossing also crosses (Corollary 1) and hence . Moreover, is not crossed between and its crossing with as is a cell. So lies on and we have , which with contradicts that and are in different components of .

Thus is an -edge and we can draw a curve from to crossing only -edges that also cross . We continue to follow along towards . As is not crossed between and (by Lemma 1 and the fact that is a -edge), we encounter a crossing of with another edge for some . Following along towards we encounter another crossing (as ) with some edge . This edge is not incident to (Lemma 1 and the fact that is a cell) and thus is incident to , making an -edge. So we can draw a fourth curve from to crossing only -edges that also cross . Finally, by Lemma 1 every edge crossing crosses every edge that crosses . Thus, there is such that does not cross any edge of . We can now introduce a new vertex to into cell with edges along , along , and along ; see Fig.5(d). The resulting drawing is still fan-planar and the new edges do not cross any edge of , as desired.

So from now on we may assume that is connected with . If is not a quadrangulation, then there exists a face whose facial walk has length at least . For each edge that intersects we have that consists of one or more segments, where for each segment either both ends are crossing points on edges of , or one end is such a crossing point and the other end is a vertex of , called a stick. If a stick has an end at vertex , its other end is crossing an edge . We call the part of containing the outer side of and the part of containing the inner side of . The inner and outer sides of sticks with an end in are defined analogously. A stick is call short if its inner side contains only two vertices (one being the stick’s end) and long, otherwise. We distinguish two cases.

Case 1. There is a long stick .

W.l.o.g. let be one end of , and let be the edge of containing the other end of , where lies on the outer side and lies on the inner side of . Let be the edge of corresponding to , and assume w.l.o.g. that no edge incident to crosses between and ; see Fig.6(a). Now we are in similar situation as in the case of a disconnected above and we can argue along the same lines.

First, if is an -edge or is a -edge, then is an edge of by Lemma 3 that can be drawn not crossing any edge in . Hence, by maximality of and we can draw a parallel copy of in the specified way, and we are done. It remains the (more elaborate) case that is a -edge and is an -edge. As above, consider a point in the plane very close to and on the -side of and the -side of ; see Fig.6(a). By fan-planarity, every edge crossing (i.e., crossing between and ) corresponds to a long stick with one end being that crosses the edge at on the inner side of . In particular, each edge crossing is an -edge, and we can draw a curve from to crossing only -edges that also cross . Similarly, every edge crossing between and is incident to and by Corollary 1 an -edge just like . Hence we can draw a curve from to crossing only -edges that also cross .

Now consider the cell of containing . Note that is completely contained in face of . Following the same argumentation as above, there is either a vertex from on and we can extend to an edge without further crossings, or there is a vertex and we can add a new vertex at position , draw edge using , edge using , and edge in a fan-planar way not crossing any edge in ; see Fig. 6(b) for one possible scenario. In both cases we are done.

(a)
(b)
(c)
Figure 7: Illustrations for the proof of Theorem 8. Edges in are drawn thick, newly added vertices and edges are drawn in red.
Case 2. All sticks are short.

Consider a vertex that is on the inner side of a stick with end in . If the edge corresponding to is a -edge, we say that vertex is blocked. If all such sticks correspond to -edges, we say that vertex is semi-free. And if there is no such stick for , we say that is free (again, blocked, semi-free, and free are defined analogously for vertices in ).

The crucial observation is that if a vertex is blocked, then its two neighbors on are free. Hence, there exists a free vertex and a free vertex and can be added crossing-free to , or three consecutive vertices in (or in ) are not blocked, in which case we can add a new vertex of degree  to ; see Fig. 6(c).

So in all cases, we can add a new edge or a new vertex with three new incident edges to , such that the resulting drawing is still fan-planar, the resulting graph is still bipartite, and new edges do not cross any edge of . ∎

Adding to an edge or a vertex with three edges, strictly increases the average degree in . Hence, we ultimatively obtain supergraphs of and of satisfying P.1–P.3. Next, we show that the charge of every original vertex is at most its degree in minus .

Claim 2.

Every satisfies .

Proof.

Without loss of generality consider any and let and be the set of edges charging . First observe that no two edges of can cross. In fact, if charges  and crosses , then charges . Now consider the face of  created by removing from the graph, and the closed facial walk around . Walk  has length exactly (counting with repetitions) as is a quadrangulation. Moreover, every edge in lies completely in and has both endpoints on . Hence, the subgraph of consisting of all edges in is crossing-free and has vertex set . Define a new graph by breaking the repetitions along the walk , i.e., consists of a cycle of length and every edge in is an uncrossed chord of this cycle. As is outerplanar and bipartite, it has at most chords. Thus, , as desired. ∎

Let be the set of newly added vertices. For each , we have and . Thus, , and by Claim 2 we get

On the other hand, by P.3 and by P.1, which together give

and conclude the proof. ∎

8 Bipartite 2-planarity

In this section we give an almost tight bound on the density of bipartite -planar graphs. We start as usual with the lower bound.

Theorem 9.

There exist infinitely many bipartite -vertex -planar (i) graphs with exactly edges, and (ii) multigraphs with exactly edges.

Proof.

For the first part, Fig.7(a) shows a construction that yields -vertex bipartite -planar graphs with edges. As in the proofs of Theorems 1 and 3, the graph is composed of quadrangular grid of size , which we wrap around a cylinder by identifying the vertices of its topmost row with the ones of its bottommost row. Thus, the two bases of the cylinder are faces of length . For each triple of consecutive faces on the same row, we can draw three edges violating neither bipartiteness nor -planarity, as in the gray shaded area in Fig.7(a). Observe that consecutive triples in the same row share one face. This implies that on average, one can draw three edges for every two faces (except for the four faces adjacent to the outermost quadrangles). Finally, we can add two additional edges inside each of the innermost and the outermost quadrangle (dashed in Fig.7(a)). Hence, the constructed graph has vertices and in total edges. For the second part, observe that if we allow non-homotopic multiedges, then we can add four additional edges; see Fig. 7(b)

(a)
(b)
Figure 8: Constructions for a bipartite -vertex: (a) -planar graph with edges, and (b) -planar multigraph with edges.

Since the proof of the upper bound is quite technical, we first give a high level description of the main steps of this proof (see Section 8.1). The details of the proof are then given later in this section (see Section 8.2).

8.1 The overview of our approach

In our proof, we first study structural properties of the planar structure of an optimal bipartite -planar graph . Let be an edge of that does not belong to . By the maximality of , edge has at least one crossing with an edge of . The part of that starts from (from ) and ends at the first intersection point of with an edge of is a stick of (of ). When has exactly two crossings, there is a part of it that is not a stick, which we call middle-part. Each part of an edge, either stick or middle-part, lies inside a face of . In this case, we say that contains this part. Let be a face of with and let be a stick of , for some , contained in . We call  a short stick, if it ends either at or at of ; otherwise, is called a long stick; see Figs. 8(a)-8(b).

(a) short stick
(b) long stick
(c) scissor + twin
(d) pseudo-scissor
Figure 9: Illustration of sticks, scissors and twins.

In the following, we will assume that among all optimal bipartite -planar graphs with  vertices, is chosen such that its planar structure is the densest among the planar structures of all other optimal bipartite -planar graphs with vertices; we call maximally dense. Under this assumption, we first prove that is a spanning quadrangulation (Lemma 5 in Section 8.2). For this, we first show that is connected (Lemma 4 in Section 8.2), as otherwise it is always possible to augment it by adding an edge joining two connected components of it. Then, we show that all faces of are of length four. Our proof by contradiction is rather technical; assuming that there is a face with length greater than four in , we consider two main cases: (i) contains no sticks, but middle-parts, and (ii) contains at least one stick. With a careful case analysis, we lead to a contradiction either to the maximality of or to the fact that is optimal.

Since is a quadrangulation, it has exactly edges and faces. Our goal is to prove that the average number of sticks for a face is at most . Since the number of edges of equals half the number of sticks over all faces of , this implies that cannot have more than edges, which gives the desired upper bound.

Let be a face of . Denote by the number of sticks contained in . A scissor of  is a pair of crossing sticks starting from non-adjacent vertices of , while a twin of is a pair of sticks starting from the same vertex of crossing the same boundary edge of ; see Fig. 8(c). We refer to a pair of crossing sticks starting from adjacent vertices of as a pseudo-scissor; see Fig. 8(d). Next, we show that a face of contains a maximum number of sticks (that is, ) only in the presence of scissors or twins, due to -planarity (see Lemma 6 in Section 8.2).

An immediate consequence of the aforementioned property is that , for every face containing a pseudo-scissor (Corollary 2 in Section 8.2). We now consider specific “neighboring” faces of a face of with four sticks and prove that they cannot contain so many sticks. Observe that each edge corresponding to a stick of starts from a vertex of and ends at a vertex of another face of . We call this other face, a neighbor of this stick. The set of neighbors of the sticks forming a scissor (twin) of form the so-called neighbors of this scissor (twin). Since , face contains two sticks and forming a twin or a scissor, with neighbors and . By -planarity and based on a technical case analysis, we show that except for a single case, called -sticks configuration and illustrated in Fig. 9(a), for which (refer to Lemmas 1011 in Section 8.2).

(a)
(b)
Figure 10: Illustration of (a) the -sticks configuration, and (b) its elimination.

Assume first that does not contain any -sticks configuration. We introduce an auxiliary graph , which we call dependency graph, having a vertex for each face of . Then, for each face of containing a scissor or a twin with neighbors and , such that , graph has an edge from to ; note that is possible. To prove that the average number of sticks for a face of is at most (which implies the desired upper bound), it suffices to prove that the number of faces of that contain two sticks is at least as large as the number of faces that contain four sticks. The latter is guaranteed by the following facts for every face of : (i) if , then has two outgoing edges and no incoming edge in , (ii) if , then the number of outgoing edges of in is at least as large as the number of its incoming edges, and finally (iii) if , then has at most two incoming edges in (see Properties 1, 2 and 3 in Section 8.2). Hence, if does not contain any -sticks configuration, then has at most edges.

To complete the proof, assume now that contains -sticks configurations. We eliminate each of them (without introducing new ones) by adding one vertex, and by replacing two edges of by six other edges violating neither its bipartiteness nor its -planarity, as in Fig. 9(b). Note that the derived graph has a planar structure that is a spanning quadrangulation not containing any -sticks configuration. Since has one vertex and four edges more than for each -sticks configuration and since the vertices of have degree at most on average, by reversing the augmentation steps we can conclude that cannot have a larger edge density than . This implies the main results of this section, that is, a bipartite -vertex -planar multigraph has at most edges (see Theorem 10 in Section 8.2).

8.2 The details of our approach

In this subsection, we give the details of our approach. We start by proving that a maximally dense planar structure of an optimal bipartite -planar graph is connected.

Lemma 4.

Let be an optimal bipartite -planar graph, such that its planar structure is maximally dense. Then, the planar structure of is connected.

Proof.

Suppose, for a contradiction, that is not connected. Since is assumed to be connected, there exists an edge in such that and belong to two different connected components and of , respectively. Note that is crossed by at least an edge of ; we assume without loss of generality that the crossing with is the first one that is encountered when walking along from to . Then, may be crossed by another edge , which may belong to or to . We assume that () does not belong to the same partition as (as ), while () does. This implies that , , , and may be added to without violating bipartiteness.

(a)
(b)
(c)
(d)
Figure 11: Augmentation of into a connected .

Suppose first that does not belong to , which implies that edge does not belong to . Consider a curve from to that first follows till its intersection point with , and then follows this edge till . Note that the first part of does not cross any edge, while the second one crosses at most one edge, call it ; see Fig. 10(a). If does not cross any edge, then we can add edge to , contradicting either the optimality of or the fact that has been chosen so that is the densest possible. If crosses , then observe that belongs to , since crosses which belongs to . If does not belong to , then we draw it as , we add it to , and we remove from . If belongs to and crosses , then we redraw as and add it to , which leads to a contradiction the fact that has been chosen so that is the densest possible. Finally, if belongs to and does not cross