On Optimal 2- and 3-Planar Graphs

On Optimal 2- and 3-Planar Graphs

Michael A. Bekos, Michael Kaufmann, Chrysanthi N. Raftopoulou

Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Germany
{bekos,mk}@informatik.uni-tuebingen.de
School of Applied Mathematical & Physical Sciences, NTUA, Greece
crisraft@mail.ntua.gr
Abstract

A graph is -planar if it can be drawn in the plane such that no edge is crossed more than times. While for , optimal -planar graphs, i.e. those with vertices and exactly edges, have been completely characterized, this has not been the case for . For and , upper bounds on the edge density have been developed for the case of simple graphs by Pach and Tóth, Pach et al. and Ackerman, which have been used to improve the well-known “Crossing Lemma”. Recently, we proved that these bounds also apply to non-simple - and -planar graphs without homotopic parallel edges and self-loops.

In this paper, we completely characterize optimal - and -planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes.

1 Introduction

Topological graphs, i.e. graphs that usually come with a representation of the edges as Jordan arcs between corresponding vertex points in the plane, form a well-established subject in the field of geometric graph theory. Besides the classical problems on crossing numbers and crossing configurations [3, 20, 26], the well-known ”Crossing Lemma” [2, 19] stands out as a prominent result. Researchers on graph drawing have followed a slightly different research direction, based on extensions of planar graphs that allow crossings in some restricted local configurations [7, 12, 14, 16, 18]. The main focus has been on 1-planar graphs, where each edge can be crossed at most once, with early results dating back to Ringel [23] and Bodendiek et al. [8]. Extensive work on generation [24], characterization [17], recognition [11], coloring [9], page number [5], etc. has led to a very good understanding of structural properties of 1-planar graphs.

Pach and Tóth [22], Pach et al. [21] and Ackerman [1] bridged the two research directions by considering the more general class of -planar graphs, where each edge is allowed to be crossed at most times. In particular, Pach and Tóth provided significant progress, as they developed techniques for upper bounds on the number of edges of simple -planar graphs, which subsequently led to upper bounds of  [22],  [21] and  [1] for simple -, - and -planar graphs, respectively. An interesting consequence was the improvement of the leading constant in the ”Crossing Lemma”. Note that for general , the current best bound on the number of edges is  [22].

Recently, we generalized the result and the bound of Pach et al. [21] to non-simple graphs, where non-homotopic parallel edges as well as non-homotopic self-loops are allowed [6]. Note that this non-simplicity extension is quite natural and not new, as for planar graphs, the density bound of still holds for such non-simple graphs.

In this paper, we now completely characterize optimal non-simple - and -planar graphs, i.e. those that achieve the bounds of and on the number of edges, respectively; refer to Theorems 1 and 2. In particular, we prove that the commonly known -planar graphs achieving the upper bound of edges, are in fact, the only optimal -planar graphs. Such graphs consist of a crossing-free subgraph where all not necessarily simple faces have size . At each face there are more edges crossing in its interior. We correspondingly show that the optimal -planar graphs have a similar simple and regular structure where each planar face has size and contains additional crossing edges.

The remainder of this paper is structured as follows: In Section 2 we introduce preliminary notions and notation. In Section 3 we present several structural properties of optimal - and -planar graphs that we use in Sections 4 and 5 in order to give their characterizations. We conclude in Section 6 with further notable insights and research directions.

2 Preliminaries

Let be a (not necessarily simple) topological graph, i.e.  is a graph drawn on the plane, so that the vertices of are distinct points in the plane, its edges are Jordan curves joining the corresponding pairs of points, and: (i) no edge passes through a vertex different from its endpoints, (ii) no edge crosses itself and (iii) no two edges meet tangentially. Let be such a drawing of . The crossing graph of has a vertex for each edge of  and two vertices of are connected by an edge if and only if the corresponding edges of cross in . A connected component of is called crossing component. Note that the set of crossing components of defines a partition of the edges of . For an edge of we denote by the crossing component of which contains .

An edge in is called a topological edge (or simply edge, if this is clear in the context). Edge is called true-planar, if it is not crossed by any other edge in . The set of all true-planar edges of forms the so-called true-planar skeleton of , which we denote by . Since is not necessarily simple, we will assume that contains neither homotopic parallel edges nor homotopic self-loops, that is, both the interior and the exterior regions defined by any self-loop or by any pair of parallel edges contain at least one vertex. For a positive integer , a cycle of length is called true-planar -cycle if it consists of true-planar edges of . If is a true-planar edge, then , while for a chord of a true-planar -cycle that has no vertices in its interior, it follows that all edges of are also chords of this -cycle. Let be a facial -cycle of with length . The order of the vertices (and subsequently the order of the edges) of is determined by a walk around the boundary of in clockwise direction. Since is not necessarily simple, a vertex or an edge may appear more than once in this order; see Figure 1. More in general, a region in is defined as a closed walk along non-intersecting segments of Jordan curves that are adjacent either at vertices or at crossing points of . The interior and the exterior of a connected region are defined as the topological regions to the right and to the left of the walk.

(a)  
(b)  
(c)  
(d)  
(e)  
(f)  
Figure 1: (a) A non-simple face , where is identified with . Different configurations used in (b–d) Lemma 1, and (e–f) Lemma 2.

Drawing is called -planar if every edge in is crossed at most times. Accordingly, a graph is called -planar if it admits a -planar drawing. An optimal -planar graph is a -planar graph with the maximum number of edges. In particular, we consider optimal - and -planar graphs achieving the best-known upper bounds of and edges. For an optimal -planar graph on vertices, a -planar drawing of is called planar-maximal crossing-minimal or simply PMCM-drawing, if and only if has the maximum number of true-planar edges among all -planar drawings of and, subject to this restriction, has also the minimum number of crossings.

Consider two edges and that cross at least twice in . Let and be two crossing points of and that appear consecutively along in this order from to (i.e., there is no other crossing point of and between and ). W.l.o.g. we can assume that and appear in this order along from to as well. In Figures 1 and 1 we have drawn two possible crossing configurations. First we drew edge as an arc with above and the edge-segment of between and to the right of . The edge-segment of between and , starts at and ends at either from the right (Figure 1) or from the left (Figure 1) of , yielding the two different crossing configurations.

Lemma 1.

For , let be a PMCM-drawing of an optimal -planar graph in which two edges and cross more than once. Let and be two consecutive crossings of and along , and let be the region defined by the walk along the edge segment of from to and the one of from to . Then, has at least one vertex in its interior and one in its exterior.

Proof.

Consider first the crossing configuration of Figure 1. Since and are consecutive along and does not cross itself, vertex lies in the exterior of , while vertex in the interior of . Hence, the lemma holds. Consider now the crossing configuration of Figure 1. Since and are consecutive along , vertices and are in the exterior of . Assume now, to the contrary, that contains no vertices in its interior. W.l.o.g. we further assume that and is a minimal crossing pair in the sense that, cannot contain another region defined by any other pair of edges that cross twice; for a counterexample see Figure 1. Let and be the number of crossings along and that are between and , respectively (red in Figure 1). Observe that by the “minimality” criterion of and we have . We redraw edges and by exchanging their segments between and and eliminate both crossings and without affecting the -planarity of ; see the dotted edges of Figure 1. This contradicts the crossing minimality of . ∎

Lemma 2.

For , let be a PMCM-drawing of an optimal -planar graph in which two edges and incident to a common vertex cross. Let be the first crossing of them starting from and let be the region defined by the walk along the edge segment of from to and the one of from to . Then, has at least one vertex in its interior and one in its exterior.

Proof.

Since is the first crossing point of and along from , vertex is not in the interior of . If , then is indeed in the exterior of . Otherwise, if and there is no other vertex in the exterior of , then is a homotopic self-loop; a contradiction. Assume now, to the contrary, that contains no vertices in its interior. W.l.o.g. we further assume that and is a minimal crossing pair in the sense that, cannot include another region defined any other pair of crossing edges incident to a common vertex; for an example see Figure 1. Denote by and the number of crossings along and that are between and , respectively (red drawn in Figure 1). First assume that . We proceed by eliminating crossing without affecting the -planarity of ; see the dotted-drawn edges of Figure 1. This contradicts the crossing minimality of . It remains to consider the case where . Assume w.l.o.g. that . By the “minimality”assumption there is an edge that crosses at least twice edge . By Lemma 1, is not an empty region; a contradiction. ∎

In our proofs by contradiction we usually deploy a strategy in which starting from an optimal - or -planar graph , we modify and its drawing by adding and removing elements (vertices or edges) without affecting its - or -planarity. Then, the number of edges in the derived graph forces to have either fewer or more edges than the ones required by optimality (contradicting the optimality or the -planarity of , resp.). To deploy the strategy, we must ensure that we do not introduce homotopic parallel edges or self-loops, and that we do not violate basic properties of (e.g., introduce a self-crossing edge). We next show how to select and draw the newly inserted elements.

A Jordan curve connecting vertex to of is called a potential edge in drawing if and only if does not cross itself and is not a homotopic self-loop in , that is, either or and there is at least one vertex in the interior and the exterior of . Note that and are not necessarily adjacent in . However, since each topological edge of is represented by a Jordan curve in , it follows that edge is by definition a potential edge of among other potential edges that possibly exist. Furthermore, we say that vertices define a potential empty cycle in , if there exist potential edges , for and potential edge of , which (i) do not cross with each other and (ii) the walk along the curves between defines a region in that has no vertices in its interior. Note that is not necessarily simple.

Lemma 3.

For , let be a PMCM-drawing of a -planar graph . Let also be a potential empty cycle of length in and assume that edges of are drawn completely in the interior of , while edges of are crossing111Note that the boundary edges of are not necessarily present in . the boundary of . Also, assume that if one focuses on of , then pairwise non-homotopic edges can be drawn as chords completely in the interior of without deviating -planarity.

  1. If , then is not optimal.

  2. If is optimal and , then all boundary edges of exist222We say that a Jordan curve exists in if and only if is homotopic to an edge in . in .

Proof.

(LABEL:*prp:nonoptim) If we could replace the edges of that are either drawn completely in the interior of or cross the boundary of with the ones that one can draw exclusively in the interior of , then the lemma would trivially follow. However, to do so we need to ensure that this operation introduces neither homotopic parallel edges nor homotopic self-loops. Since the edges that we introduce are potential edges, it follows that no homotopic self-loops are introduced. We claim that homotopic parallel edges are not introduced either. In fact, if and are two homotopic parallel edges, then both must be drawn completely in the interior of , which implies that and are both newly-introduced edges; a contradiction, since we introduce pairwise non-homotopic edges. (LABEL:*prp:boundary) In the exchanging scheme that we just described, we drew edges as chords exclusively in the interior of . Of course, one can also draw the boundary edges of , as long as they do not already exist in . Since is optimal, these edges must exist in . ∎

(a)  
(b)  
(c)  
(d)  
Figure 2: (a–c) A potential empty cycle with (a)  and five chords with two crossings each, (b)  and six chords with at most two crossings each, and (c)  and eight chords with at most three crossings each. (d) Configuration used in the proof of Property 2.

Note that in Lemma 3 the edges that are drawn completely in the interior of the potential empty cycle and the edges that cross its boundary, are the only edges that have at least one edge-segment within . This means that we can compute by counting the edges that have at least one edge-segment within . In the following sections, there will be some standard cases where we apply Lemma 3. In most of them, a potential empty cycle on five or six vertices is involved, that is, . If , then one can draw five chords in the interior of without affecting its - or -planarity; see Figure 2. If , then one can draw either six or eight chords in the interior of without affecting its - or -planarity, respectively; see Figures 2 and 2.

3 Properties of optimal 2- and 3-planar graphs

In this section, we investigate properties of optimal - and -planar graphs.We prove that a PMCM-drawing of an optimal - or -planar graph can contain neither true-planar cycles of a certain length nor a pair of edges that cross twice. We use these properties to show that is quasi-planar, i.e. it contains no pairwise crossing edges. First, we give the following definition. Let be a simple closed region that contains at least one vertex of in its interior and one in its exterior. Let () be the subgraph of whose vertices and edges are drawn entirely in the interior (exterior) of . Note that () is not necessarily an induced subgraph of , since there could be edges that exit and enter . We refer to and as the compact subgraphs of defined by . The following lemma, used in the proofs for several properties of optimal - and -planar graphs, bounds the number of edges in any compact subgraph of .

Property 1.

Let be a drawing of an optimal - or -planar graph and let be a compact subgraph of on vertices that is defined by a closed region . If , has at most edges if is optimal -planar, and at most edges if is optimal -planar. Furthermore, there exists at least one edge of crossing the boundary of in .

Proof.

We prove this property for the class of -planar graphs; the proof for the class of -planar graphs is analogous. So, let be a drawing of an optimal -planar graph with vertices and edges. Let and be two compact subgraphs of defined by a closed region . For let and be the number of vertices and edges of . Suppose that . In the absence of , drawing might contain homotopic parallel edges or self-loops. To overcome this problem, we subdivide an edge-segment of the unbounded region of by adding one vertex.333One can view this process as replacing with a single vertex; thus no homotopic parallel edges exist in . Then we move this vertex towards the edge-segment we want to subdivide until it touches it. The derived graph, say , has vertices and edges. Since has no homotopic parallel edges or self-loops and , it follows that , which gives .

For the second part, assume for the sake of contradiction that no edge of crosses the boundary of . This implies that . We consider first the case where . By the above we have that and . Since and , it follows that ; a contradiction to the optimality of . Since a graph consisting only of two non-adjacent vertices cannot be optimal, it remains to consider the case where either or . W.l.o.g. assume that . Since , it follows that , which implies ; a contradiction to the optimality of . ∎

For two compact subgraphs and defined by a closed region , Property 1 implies that the drawings of and cannot be “separable”. In other words, either there exists an edge connecting a vertex of with a vertex of , or there exists a pair of edges, one connecting vertices of and the other vertices of , that cross in the drawing .

Property 2.

In a PMCM-drawing of an optimal -planar graph there is no empty true-planar cycle of length three.

Proof.

Assume to the contrary that there exists an empty true-planar -cycle in on vertices , and . Since is connected and since all edges of are true-planar, there is neither a vertex nor an edge-segment in , i.e., is a chordless facial cycle of . This allows us to add a vertex in its interior and connect to vertex by a true-planar edge. Now vertices , , , and define a potential empty cycle of length five, and we can draw five chords in its interior without violating -planarity and without introducing homotopic parallel edges or self-loops; refer to Figure 2. The derived graph has one more vertex than and six more edges. Hence, if and are the number of vertices and edges of respectively, then has vertices and edges. Then , which implies that has more edges than allowed; a contradiction. ∎

Property 3.

The number of vertices of an optimal -planar graph is even.

Proof.

Follows directly from the density bound of of . ∎

Property 4.

A PMCM-drawing of an optimal -planar graph has no true-planar cycle of odd length.

Proof.

Let be an odd number and assume to the contrary that there exists a true-planar -cycle in . Denote by (, respectively) the subgraph of induced by the vertices of and the vertices of that are in the interior (exterior, respectively) of in without the chords of that are in the exterior (interior, respectively) of in . For , observe that contains a copy of . Let and be the number of vertices and edges of that do not belong to . Based on graph , we construct graph by employing two copies of that share cycle . Observe that is -planar, because one copy of can be embedded in the interior of , while the other one in its exterior. Hence, in this embedding, there exist neither homotopic self-loops nor homotopic parallel edges. Let and be the number of vertices and edges of that do not belong to . If has vertices and edges, then by construction the following equalities hold: (i) , (ii) , (iii) , and (iv) .

We now claim that . When the claim clearly holds. Otherwise (i.e., ), cycle is degenerated to a self-loop which must contain at least one vertex in its interior and its exterior. Hence, the claim follows. Property 3 in conjunction with Eq.(i) implies that is not optimal, that is, . Hence, by Eq.(ii) it follows that . Summing up over , we obtain that . Finally, from Eq.(iii) and Eq.(iv) we conclude that ; a contradiction to the optimality of . ∎

Property 5.

In a PMCM-drawing of an optimal -planar graph there is no pair of edges that cross twice with each other.

Proof.

Assume to the contrary that and cross twice in at points and . By -planarity no other edge of crosses and . Let be the region defined by the walk along the edge segment of between and and the edge segment of between and . As mentioned in the proof of Lemma 1, there exist two crossing configurations for and ; see Figures 1 and 1. In the crossing configuration of Figure 1, vertices and are in the interior of , while vertices and in its exterior. Hence, and hold. We redraw and by exchanging the middle segments between and and eliminate both crossings and without affecting -planarity; see the dotted edges of Figure 1. Note that since and the two edges cannot be homotopic self-loops. Also, no homotopic parallel edges are introduced, since this would imply that at least one of the two edges already exists in violating -planarity. Now consider the crossing configuration of Figure 1. By Lemma 1, has at least one vertex in its interior. By -planarity we have that no edge of crosses the boundary of ; a contradiction to Property 1. ∎

Property 6.

In a PMCM-drawing of an optimal -planar graph there is no pair of edges that cross more than once with each other.

Proof.

We have already noted that a pair of edges cannot cross more than twice in . Assume to the contrary that two edges and of cross (exactly) twice in . Figures 3 and 3 illustrate the two possible different crossing configurations. Let and be their crossing points. By Lemma 1 it follows that the region that is defined by the walk along the the edge segment of between and and the edge segment of between and has at least one vertex in its interior. Let be the subgraph of that is drawn completely in the interior of in . By -planarity, there exist at most two edges and that cross and respectively.

In both crossing configurations we proceed to define two Jordan curves and in with endpoints and , so that their union contains only in its interior the vertices of ; see Figures 3 and 3. Curve emanates from vertex , follows edge up to point and ends at vertex by following edge . Curve emanates from vertex , follows edge up to point , follows edge up to point , follows edge up to point and ends at vertex by following edge .

We now claim that both curves and are potential edges. By definition, our claim holds when . Assume now that . Let be the region defined by the walk along the edge-segment of from to and the edge-segment of from to (where ). By Lemma 2 has at least one vertex in its interior and at least one vertex in its exterior. This implies that the first of our curves, i.e. , which encloses region is a potential edge.

Now, assume to the contrary that is not a potential edge. Then . Let be the region defined by the walk along the edge-segment of from to , the edge-segment of from to , the edge-segment of from to and the edge-segment of from to (where ). Since lies in the interior of and is not a potential edge, region has no vertices in its exterior; refer to Figure 3. Note that in Figure 3 we illustrate the same case assuming . By Property 4 potential edge must be crossed (as otherwise it is a true-planar self-loop in ). This implies that there exists at least one edge that crosses . This edge must also cross or and is therefore either edge or edge . Suppose w.l.o.g. that is edge ; see Figure 3. Let be the crossing point of and . Now edge has exactly three crossings. We redraw and by exchanging their edge-segments between their common endpoint and their first crossing , so as to eliminate . Let and be the new curves in . Since is crossing minimal, it follows that at least one of or must be homotopic parallel to an existing edge in . Since has already three crossings in , potential edge cannot exist in , as otherwise it would introduce a fourth crossing on . Hence, potential edge must exist in and this is edge . Now we focus on edge . Edge has an endpoint in the interior of and crosses . However, since has no vertices in its exterior, and edges and have already three crossings, edge must end at vertex . In this case, edges and have as a common endpoint and cross at point . Hence, region defined by the walk along the edge segment of from to and the edge segment of from to contains at least one vertex in its interior. However, is contained in the exterior of , and therefore there exists at least one vertex in the exterior of , which is a contradiction. Hence, is a potential edge.

(a)  
(b)  
(c)  
(d)  
Figure 3: Configurations used in Property 6.

We proceed by removing from all vertices and edges of , edges , , as well as the edge that crosses , if any. Then, the cycle formed by potential edges and becomes empty and this allows us to follow an approach similar to the one described in the proof of Lemma 3. More precisely, we add in the interior of this potential empty cycle two vertices and , such that , and form a path (in this order) that is completely drawn in its interior. The union of this path with and defines in the derived drawing a new (non-simple) potential empty cycle of length six. In its interior one can embed additional edges as in Figure 2. Summarizing, if has vertices and edges, we removed from exactly vertices and at most edges and this allowed us to introduce two new vertices and edges without affecting -planarity. Let be the derived -planar graph. The fact that contains neither homotopic parallel edges nor homotopic self-loops can be argued as in the proof of Lemma 3.(i). If has vertices and edges, then has vertices and edges, where edges. We distinguish two cases depending on whether has one or more vertices. If , then . Also, has exactly one more vertex than . Since is optimal, by Property 3 it follows that cannot be optimal. Hence, , which implies that ; a contradiction to the density of . On the other hand if , by Property 1 we have that , as is a compact subgraph of defined by . This gives , that is has more edges than allowed; a clear contradiction. ∎

Now assume that contains three mutually crossing edges , and . In Figures 44 we have drawn four possible crossing configurations. First, we drew and w.l.o.g. as vertical and horizontal line-segments that cross at point . Then, we placed vertex and drew the first segment of its edge crossing w.l.o.g. the edge-segment of between and at point from above. So the middle segment of starts at and has to end at edge , either from left or right, and either in the lower or in the upper segment. This gives rise to the four configurations demonstrated in Figures 44, which we examine in more details in the following. Note that the endpoints of the three edges are not necessarily distinct (e.g., in Figure 4 we illustrate the case where and for the crossing configuration of Figure 4). For each crossing configuration, one can draw curves connecting the endpoints of , and (red colored in Figures 44), which define a region that has no vertices in its interior. This region fully surrounds and and the two segments of that are incident to vertices and .

(a)  
(b)  
(c)  
(d)  
(e)  
Figure 4: Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.
Claim 1.

Each of the crossing configurations of Figures 4-4 induces at least potential edges.

Proof.

Observe that all solid-drawn red curves of Figures 44 are indeed potential edges: If for example of Figure 4 is not a potential edge, then and is a self-loop with no vertex either in its interior or in its exterior; a contradiction to Lemma 2. ∎

Claim 2.

The crossing configuration of Figure 4 induces at least four potential edges.

Proof.

As in Claim 1 we can prove that , , and are potential edges. ∎

Corollary 1.

The configuration of Figure 4 induces a potential empty cycle  of length . Each of the configurations of Figures 44 induces a potential empty cycle  of length .

Claim 3.

In the case where the crossing configuration of Figure 4 induces exactly four potential edges, there exists at least one vertex in the interior of region defined by the walk along the edge segment of between and , the edge segment of between and and the edge segment of between and .

Proof.

By Claim 2, , and must be homotopic self-loops; see Figure 4. In this case, edges and are incident to a common vertex, namely and cross. By Lemma 2 region (red-shaded in Figure 4) has at least one vertex in its interior. Since is the union of the interior of and the homotopic self-loop , contains at least one vertex in its interior. ∎

Property 7.

A PMCM-drawing of an optimal -planar graph is quasi-planar.

Proof.

Assume to the contrary that there exist three mutually crossing edges , and in ; see Figure 4. By Corollary 1, there is a potential empty cycle of length at least . By -planarity, there is no other edge crossing , or . Hence, the only edges that are drawn in the interior of are and , while is the only edge that crosses the boundary of .

First, consider the case where is of length . Since we can draw at least five chords completely in the interior of as in Figure 2 or 2 without violating its -planarity, it follows by Lemma 3.(i) (for and ) that is not optimal; a contradiction. Finally, consider the case where is of length four. In this case, we have the crossing configuration of Figure 4. By Claim 3 there is at least one vertex in the interior of region . More in general, let be the compact subgraph of that is completely drawn in the interior of region . Since edges , and have already two crossings, it follows that no edge of crosses the boundary of ; a contradiction to Property 1. ∎

Property 8.

A PMCM-drawing of an optimal -planar graph is quasi-planar.

Proof.

As in the case of -planar optimal graphs, assume that there exist three mutually crossing edges , and in . By Corollary 1, there is always a potential empty cycle of length at least . Since , and have already two crossings each, there exist at most three other edges that cross , or . Hence, the only edges that are drawn in the interior of are and , while and at most three other edges of cross the boundary of . We distinguish three cases depending on whether has length , or .

Consider first the case where has length six. Since we can draw eight chords completely in the interior of as in Figure 2 without deviating -planarity, it follows by Lemma 3.(i) (for and ) that is not optimal; a contradiction.

Consider now the case where has length five. We claim that at least one boundary edge of does not exist in . In order to prove the claim, we consider the four crossing configurations of Figure 5 separately. In Figure 5, if potential edge is an edge in , then it crosses twice , contradicting Property 6. For Figures 55, if all red drawn curves belong to , then crosses , and at least two of the boundary edges of , violating -planarity. Hence, our claim follows. We proceed by removing edges , and and any other edge crossing the boundary of from , and we add five chords in the interior of , along with one “missing” boundary edge of . Let be the derived graph. Note that, we removed at most six edges and added at least six. This implies that is also optimal. However, is a true-planar -cycle in the drawing of , contradicting Property 4.

(a)  
(b)  
(c)  
(d)  
(e)  
(f)  
Figure 5: Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.

It remains to consider the case where is of length four. By Claim 3 there is at least one vertex in the interior of region . As in the proof of Property 7, we denote by the subgraph of completely drawn in region . is a compact subgraph of and by Property 1, it follows that if has vertices, then it has edges (note that if , then ). We replace with one vertex, say , we keep edges , and and remove any edge crossing , or in . We redraw the edge-segment of incident to so as to be incident to (without introducing new crossings). Finally, we add edges , and ; see Figure 5. The derived graph has vertices and at least edges, where and are the number of vertices and edges of . For , we have that , i.e., has more edges than allowed. In the case where and , it follows that has the same number of edges as and is therefore optimal. However, potential edges , and can be added in (if not present) forming thus a true-planar -cycle; a contradiction to Property 4. ∎

We next present a refinement of the notion of potential edges. In particular, we focus on two main categories of potential edges that we will heavily use in Sections 4 and 5. Consider a pair of vertices and of that are not necessarily distinct. We say that and form a corner pair if and only if an edge crosses an edge for some and in ; see Figure 6. Let be the crossing point of and . Then, any Jordan curve joining vertices and induces a region that is defined by the walk along the edge-segment of from to , the edge segment of from to and the curve from to . We call corner edge with respect to  and if and only if has no vertices of in its interior.

(a)  
(b)  
(c)  
(d)  
(e)  
Figure 6: (a-b) vertices and form a corner pair; (c-d) vertices and form a side pair; (e) at least one of the two potential side-edges exists.
Property 9.

In a PMCM-drawing of an optimal -planar graph any corner edge is a potential edge.

Proof.

By the definition of potential edges, the property holds when . Consider now the case where . In this case is a self-loop; see Figure 6. If the property does not hold, then it follows that is a self-loop with no vertices either in its interior or in its exterior. However, this contradicts Lemma 2, and the property holds. ∎

We say that vertices and form a side pair if and only if there exist edges and for some and such that they both cross a third edge in and additionally ; see Figure 6 or 6. Let and be the crossing points of and with , respectively. Assume w.l.o.g. that and appear in this order along from vertex to vertex . Also assume that the edge-segment of between and is on the same side of edge as the edge-segment of between and ; refer to Figure 6. Then, any Jordan curve joining vertices and induces a region that is defined by the walk along the edge-segment of from to , the edge segment of from to , the edge segment of from to and the curve from to . We call side-edge w.r.t.  and if and only if has no vertices of in its interior. Since by Properties 7 and 8 edges and cannot cross with each other (as they both cross ), it follows that region is well-defined. Symmetrically we define region and side-edge with respect to  and .

Property 10.

In a PMCM-drawing of an optimal -planar graph with at least one of the side-edges , is a potential edge.

Proof.

Before giving the proof, note that since edges , and do not mutually cross, curves and cannot cross themselves. Now, for a proof by contradiction, assume that neither nor are potential edges. This implies that , and both and are self-loops that have no vertices in their interiors or their exteriors. Figure 6 illustrates the case where both and are self-loops with no vertices in their interiors; the other cases are similar. It is not hard to see that and are homotopic side-edges; a contradiction. ∎

We say that and are side-apart if and only if both side-edges and are potential edges.

4 Characterization of optimal 2-planar graphs

By using the properties we proved in Section 3, in this section we examine some more structural properties of optimal -planar graphs in order to derive their characterization (see Theorem 1).

Lemma 4.

Let be a PMCM-drawing of an optimal -planar graph . Any edge that is crossed twice in is a chord of a true-planar -cycle in .

Proof.

Let be an edge of that is crossed twice in by edges and at points and , respectively. Note that, by Property 5 edges