2 -Layer k -Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth
Patrizio Angelini, Giordano Da Lozzo, Henry Förster, Thomas Schneck
22-Layer k -Planar Graphs Density, Crossing Lemma, Relationships, and Pathwidth
Patrizio Angelini − − − , Giordano Da Lozzo − − − ,Henry F¨orster − − − , and Thomas Schneck − − − John Cabot University, Rome, Italy [email protected] Roma Tre University, Rome, Italy [email protected] University of T¨ubingen, T¨ubingen, Germany { foersth,schneck } @informatik.uni-tuebingen.de Abstract.
The 2-layer drawing model is a well-established paradigm tovisualize bipartite graphs. Several beyond-planar graph classes have beenstudied under this model. Surprisingly, however, the fundamental class of k -planar graphs has been considered only for k = 1 in this context. Weprovide several contributions that address this gap in the literature. First,we show tight density bounds for the classes of 2-layer k -planar graphswith k ∈ { , , , } . Based on these results, we provide a Crossing Lemmafor 2-layer k -planar graphs, which then implies a general density bound for2-layer k -planar graphs. We prove this bound to be almost optimal with acorresponding lower bound construction. Finally, we study relationshipsbetween k -planarity and h -quasiplanarity in the 2-layer model and showthat 2-layer k -planar graphs have pathwidth at most k + 1. Keywords: · k -planar graphs · density · CrossingLemma · pathwidth · quasiplanar graphs Beyond-planarity is an active research area that studies graphs admitting drawingsthat avoid certain forbidden crossing configurations. Research on this subject hasattracted considerable interest due to its theoretical appeal and due to the needof visualizing real-world non-planar graphs. A great deal of attention has beencaptured by two important graph families. The k -planar graphs, with k ≥
1, forwhich the forbidden configuration is an edge crossing more than k other edges,and the h -quasiplanar graphs, with h ≥
3, for which the forbidden configurationis a set of h pairwise crossing edges. The study of these two families finds itsorigins in the 1960’s [11,40], when the question arose about the density of thesegraphs, that is, the maximum number of edges of graphs in these families.Many works have addressed this extremal graph theoretical question andestablished upper bounds for k -planar and h -quasiplanar graphs for various valuesof k and h . For small k and h , these upper bounds have been proven to be tightby lower bound constructions achieving the corresponding density. The mostsignificant results include tight density bounds for 1-planar graphs [39] (4 n − a r X i v : . [ c s . D M ] A ug dges), 2-planar graphs [39] (5 n −
10 edges), 3-planar graphs [13,36] (5 . n − n − k , the currently best upper boundis 3 . √ k n , which can be derived from the result of Ackerman [2] on 4-planargraphs and from the renowned Crossing Lemma [5]. For h -quasiplanar graphs,despite considerable research efforts, a density upper bound that is linear in thenumber of vertices exists only for h ≤ quasiplanar ) graphs. Here, simple means that any two edges meet in at most one point, which is either a commonendvertex or an internal point. For general h , only super-linear upper boundsare known [18,29,30,38,43,44] while a linear bound has been conjectured [38].These two families have also been studied from other perspectives. A notable re-lationship is that every simple k -planar graph is also simple ( k +1)-quasiplanar [7],for every k ≥
2. It is also known that every optimal . n − k -planar graphs is NP-complete for every k ≥ h -quasiplanar graphs is still open for any h ≥ fan-planar graphs, in which noedge is crossed by two independent edges or by two adjacent edges from differentdirections [12,16,17,33], and the RAC graphs , in which the edges are poly-lineswith few bends and crossings only happen at right angles [8,23,24,27]. Theseand other graph classes have been also investigated with respect to their density,recognition, and relationship with other classes; see also the recent survey [25].Beyond-planar classes have also been studied under additional constraintson the placement of the vertices. In the outer model [10,12,19,20,22,31,32] everyvertex is incident to the unbounded region of the drawing, while in the 2 -layermodel [16,17,21,22] the vertices lie on two horizontal lines and every edge is a y -monotone curve. The latter model requires the graph to be bipartite, and theconstraints on the placement of the vertices emphasize the bipartite structure.Beyond-planar bipartite graphs have also been considered in the general drawingmodel, without any additional restriction [9]. We remark that the 2-layer modellies at the core of the Sugiyama framework for general layered drawings [41,42].In [21], it was shown that 2-layer RAC graphs have at most n − n − h -quasiplanar graphs, Walczak [45] provided a density upper boundof ( h − n −
1) edges, following from the fact that convex bipartite geometric h -quasiplanar graphs can be ( h − n − n − K ,n is 2-layer fan-planar. Note that 2-layer fan-planar graphs have beencharacterized [16] and can be recognized when the graph is biconnected [16] or atree [15]. Another property that has been investigated in the 2-layer model isthe pathwidth. Namely, 2-layer fan-planar graphs have pathwidth 2 [15], while2-layer graphs with at most c crossings in total have pathwidth 2 c + 1 [26]; notethat both results can be extended to general layered graphs. Our Contribution.
From the above discussion it is evident that, in the wideliterature on the 2-layer model, the study of the central class of k -planar graphsis completely missing, except for the special case k = 1. In this paper, we makeseveral contributions towards filling this gap. We provide tight density boundsfor 2-layer k -planar graphs with k ∈ { , , , } in Section 3. Exploiting thesebounds, we deduce a Crossing Lemma for 2-layer graphs in Section 4. This impliesa density upper bound for general values of k . We then show a lower boundconstruction that is within a factor of 1 / .
84 from the upper bound. Finally, inSection 5, we investigate two additional properties. First, we prove that 2-layer2-planar graphs are 2-layer quasiplanar, as in the case where the vertices arenot restricted to two layers [7]. For larger k , we show a stronger relationship,namely, every 2-layer k -planar graph is 2-layer h -quasiplanar for h = (cid:6) k + 2 (cid:7) .Second, we demonstrate that 2-layer k -planar graphs have pathwidth at most k + 1, which is the first result of this type, since they may have a linear numberof crossings and may not be fan-planar. The -layer model. A bipartite graph G = ( U ˙ ∪ V, E ) is a graph with vertexsubsets U and V , so that E ⊆ U × V . A topological -layer graph is a bipartitegraph drawn in the plane so that the vertices in U and V are mapped to distinctpoints on two horizontal lines L u and L v , respectively, and the edges are mappedto y -monotone Jordan arcs. A topological 2-layer graph can be assumed to besimple, that is, no two adjacent edges cross each other, and every two independentedges cross each other at most once.Let G be a topological 2-layer graph. We denote the vertices in U and in V as u , . . . , u p and v , . . . , v q , respectively, in the order in which they appear inpositive x -direction along L u and L v . We denote the number of vertices of G by n = p + q and the number of edges in E by m . We call G k -planar if eachedge is crossed at most k times, and h -quasiplanar if there is no set of h pairwisecrossing edges. Further, we say that a bipartite graph G is 2 -layer k -planar ( h -quasiplanar ) if there exists a topological 2-layer k -planar (resp. h -quasiplanar)graph whose underlying abstract graph is isomorphic to G .The maximum number of edges of a graph class C is a function m C : N → N such that (i) every n -vertex graph in C has at most m C ( n ) edges, and (ii) forevery n , there is an n -vertex graph in C with m C ( n ) edges. The (maximumedge) density of C is a function d C : N → N such that (i) for every n , it holdshat d C ( n ) ≥ m C ( n ), and (ii) there are infinitely many values of n such that d C ( n ) = m C ( n ). We say that an n -vertex graph in C with d C ( n ) edges is optimal .Note that 2-layer quasiplanar graphs are equivalent to the convex bipartitegeometric quasiplanar graphs , where vertices lie on a convex shape so that the twopartition sets are well-separated [45]. Since these graphs are planar bipartite, asdiscussed in Section 1, and include K ,n , their density can be established using thesame argumentation as for convex bipartite geometric quasiplanar graphs in [45]: Theorem 1. An n -vertex -layer quasiplanar graph has at most n − edgesfor n ≥ . Also, there exist infinitely many -layer quasiplanar graphs with n vertices and n − edges.Tree and path decomposition. A tree decomposition of a graph G = ( V, E ) is atree T on vertices B , . . . , B n called bags such that the following properties hold:(P.1) each bag B i is a subset of V , (P.2) V = (cid:83) ni =1 B i , (P.3) for every edge ( u, v ) ∈ E , there exists a bag B i such that u, v ∈ B i , and (P.4) for every vertex v , thebags containing v induce a connected subtree of T . If T is a path, we call T a pathdecomposition . The width of a tree decomposition T is the maximum cardinality ofany of its bags minus one, i.e., width( T ) = max i ∈{ ,...,n } ( | B i | − treewidth of a graph G is the minimum width of any of its tree decompositions, whereasthe pathwidth of G is the minimum width of any of its path decompositions. k In this section, we establish the density of 2-layer k -planar graphs for smallvalues of k . We start with a preliminary observation, which follows from the factthat the density of k -planar graphs can be upper bounded by a linear functionin n [2,39] and that the density of 2-layer 1-planar graphs is lower bounded by n − Lemma 1.
For k ≥ , there exist positive rational numbers a k ≥ and b k ≥ such that (i) every n -vertex -layer k -planar graph has at most a k n − b k edgesfor n ≥ n k with n k a constant, and (ii) there is a -layer k -planar graph with n vertices and exactly a k n − b k edges for some n > . We then define a useful concept for the analysis of 2-layer k -planar graphs: Definition 1.
Let G be a topological -layer k -planar graph and let G [ i, j | x, y ] ,with ≤ i ≤ j ≤ p and ≤ x ≤ y ≤ q , be the topological subgraph of G induced byvertices { u i , . . . , u j , v x , . . . , v y } . G [ i, j | x, y ] is a brick if it contains two distinctcrossing-free edges, namely ( u i , v x ) and ( u j , v y ) , that are also crossing-free in G . The smallest brick, called trivial , contains one vertex of one partition set, say u i = u j , and two consecutive vertices of the second one, say v x and v y = v x +1 . Observation 1.
Every optimal topological -layer k -planar graph contains pla-nar edges ( u , v ) and ( u p , v q ) , and hence at least one brick. · ·· · · · · · (a) · · ·· · · · · · (b) Fig. 1: (a) A maximal topological 2-layer 2-planar graph that is not optimal, asshown by the graph in (b). Differences between the two graphs are dashed blue.Regarding the connectivity we observe the following. If a topological 2-layer k -planar graph G is not connected, we can draw the connected components asconsecutive bricks and connect two consecutive bricks with another edge. Hence,we conclude the following: Observation 2.
Every optimal topological -layer k -planar graph is connected. Next, we establish a useful property of an optimal 2-layer k -planar graph G . Lemma 2.
Let G be an optimal topological -layer k -planar graph with exactly a k n − b k edges. Then G contains no vertex of degree and no trivial brick.Proof. Assume that G contains a degree-1 vertex v and consider the graph G (cid:48) obtained from G by removing v . This graph has m (cid:48) = m − n (cid:48) = n − m (cid:48) = a k n − b k − a k ( n − − b k + ( a k − a k ( n − − b k since a k ≥ , by Lemma 1; a contradiction.Second, assume that G contains a trivial brick G [ i, i | x, x + 1]. Then, considerthe graph G (cid:48) obtained from G by identifying vertices v x and v x +1 . Clearly G (cid:48) has m (cid:48) = m − u i , v x ) and ( u i , v x +1 ) coincide in G (cid:48) ) and n (cid:48) = n − (cid:117)(cid:116) We start with an observation about maximal topological 2-layer 2-planar graphs,that is, in which no edge may be inserted without violating 2-planarity.
Observation 3.
There exists a maximal topological -layer -planar graph thatis not optimal; see Fig. 1. We now characterize the structure of bricks in optimal 2-layer 2-planar graphs.
Lemma 3.
Let G be an optimal topological -layer -planar graph with exactly a n − b edges and let G [ i, j | x, y ] be a brick of G . Then, j ≥ i + 1 and y = x + 1 ,or j = i + 1 and y ≥ x + 1 .Proof. By Lemma 2, G [ i, j | x, y ] is not a trivial brick. Assume, for a contradiction,that both y ≥ x + 2 and j ≥ i + 2. We first observe that u i is connected tosome v t (cid:54) = v x , while v x is connected to some u s (cid:54) = u i . If this were not the case,say if u i were only incident to v x , then a crossing-free edge ( v x , u i +1 ) could be i v x u i +1 · · · · · · u s v y u j · · ·· · · v t (a) u i v x u i +1 v x +1 · · ·· · · · · ·· · · u s v t v y u j (b) u i v x u i +1 v x +1 · · ·· · · · · ·· · · u s v t v y u j (c) u i v x u i +1 · · · u s v t v t (cid:48) · · · · · · v y u j u s (cid:48) · · · u i v x u i +1 · · · u s v t v t (cid:48) · · · · · · v y u j u s (cid:48) · · · (d) u i v x u i +1 u s v t v t (cid:48) · · ·· · · v y u j u i v x u i +1 u s v t v t (cid:48) · · ·· · · v y u j (e) Fig. 2: Illustrations for the proof of Lemma 3.inserted, contradicting the optimality of G ; see Fig. 2a and recall that a brick hasno crossing-free edge, except for ( u i , v x ) and ( u j , v y ). So in the following assumethat ( u i , v t ) and ( v x , u s ) belong to G [ i, j | x, y ], with v t (cid:54) = v x and u s (cid:54) = u i , suchthat there exists no edge ( u i , v t (cid:48) ) with t (cid:48) > t and no edge ( v x , v s (cid:48) ) with s (cid:48) > s .Next, we consider u i +1 and v x +1 . Assume first that u i +1 (cid:54) = u s and that v x +1 (cid:54) = v t .Then, all edges incident to u i +1 and v x +1 have a crossing with ( u i , v t ) or ( v x , u s ).Since ( u i , v t ) and ( v x , u s ) cross each other, there can be at most two such edges,and thus u i +1 or v x +1 has degree one; see Figs. 2b and 2c. By Lemma 2, thiscontradicts the optimality of G . Hence, assume w.l.o.g. that v x +1 = v t . Notethat u s (cid:54) = u i +1 , as otherwise the crossing-free edge ( u i +1 , v x +1 ) could be inserted,contradicting the optimality of G . In addition, u s = u i +2 , since otherwise u i +1 and u i +2 could only be incident to a total of two edges, by the same argumentas above, resulting in a degree-1 vertex, which contradicts the optimality of G .By Lemma 2, both u i +1 and v x +1 have degree at least 2. Let u s (cid:48) and v t (cid:48) denotethe neighbors of v x +1 and u i +1 respectively, such that s (cid:48) and t (cid:48) are maximal. Firstassume that t (cid:48) (cid:54) = t . If s (cid:48) = i + 1, the crossing-free edge ( u s , v t ) can be inserted,contradicting the optimality of G . We observe that edge ( u i +1 , v t (cid:48) ) is crossedby edges ( v x , u s ) and ( v t , u s (cid:48) ). If u s (cid:54) = u (cid:48) s , we can obtain a topological 2-layer2-planar graph G (cid:48) by removing edge ( v x , u s ) and inserting edges ( v t , u s ) and( v t , u i +1 ); see Fig. 2d. This clearly contradicts the optimality of G . If u s = u (cid:48) s ,we can obtain a topological 2-layer 2-planar graph G (cid:48) by removing edge ( u i , v t )and inserting edges ( v x , u i +1 ) and ( v t , u i +1 ); see Fig. 2e. This again contradictsthe optimality of G . We conclude that t (cid:48) = t .Since ( v x , u s ) is crossed by edges ( u i , v t ) and ( u i +1 , v t ), we conclude that( u s , v t ) can be inserted without crossings, contradicting the optimality of G . (cid:117)(cid:116) By Lemmas 2 and 3, we get that every brick must be a K ,h for some h ≥ h ≤
3; see also Fig. 3a:
Observation 4.
The complete bipartite graph K , is not -layer -planar. We are ready to prove a tight bound for the density of 2-layer 2-planar graphs: i u j v x v x +1 (a) u i u i +1 v x v x +1 (b) u i u i +1 v x +1 v x +2 v x (c) b b b β · · · u v · · ·· · · u p v q (d) Fig. 3: The unique 2-layer drawings of (a) K , ; (b) K , ; (c) K , . (d) An optimal2-layer 2-planar graph is a sequence of bricks joint at planar edges. · · ·· · ·· · · u p v p u v Fig. 4: A family of 3-planar graphs on n = 2 p vertices with 2 n − Theorem 2.
Any -layer -planar graph on n vertices has at most n − edges.Moreover, the optimal -layer -planar graphs with exactly n − edges aresequences of K , ’s such that consecutive K , ’s share one planar edge.Proof. Lemmas 2 and 3, and Observation 4 imply that G contains only K , - and K , -bricks; see Figs. 3b and 3c. Moreover, the planar edges separate G into asequence of β bricks ( b , . . . , b β ) such that b i and b i +1 share one planar edge. Let β denote the number of K , -bricks. Then, G has β − β K , -bricks. Moreover, n = 2 β + 2 + ( β − β ) = 3 β − β + 2 since each of the β + 1 planar edges is incidentto two distinct vertices while each K , -brick contains an additional vertex; seeFig. 3c. Finally, m = β + 1 + 2 β + 4( β − β ) = 5 β − β + 1 since every K , -brickcontains two non-planar edges while every K , -brick contains four. For a fixedvalue of n , β = n + β − and the density is m = n − β − . This isclearly maximized for β = 0. Hence, the maximum density is m = n − whichis tightly achieved for graphs in which every brick is a K , . (cid:117)(cid:116) Next, we give a tight bound on the density of 2-layer 3-planar graphs. We firstpresent a lower bound construction:
Theorem 3.
There exist infinitely many -layer -planar graphs with n verticesand n − edges.Proof. We describe a family of graphs where p = q ; refer to Fig. 4. Each graphhas the following edges: ( u i , v i ) for 1 ≤ i ≤ p (red edges in Fig. 4); ( u i , v i +1 ) for1 ≤ i ≤ p −
1, and ( u i , v i − ) for 2 ≤ i ≤ p (green edges in Fig. 4); ( u i , v i +2 ) for1 ≤ i ≤ p − u , u p − , v and v p havedegree 3, u p and v have degree 2, and all other vertices have degree 4, yielding4 n − n − (cid:117)(cid:116) · · u p v p u v · · ·· · · (a) u i v x u i +2 v x +1 u i +1 v x +2 G G u p v q u v (b) Fig. 5: (a) A family of 4-planar graphs on n = 2 p vertices with 2 n − G and G .The following theorem provides the corresponding density upper bound: Theorem 4.
Let G be a topological -layer -planar graph on n vertices. Then G has at most n − edges for n ≥ . Moreover, if G is optimal, it is quasiplanar.Proof (Sketch). We show that optimal 2-layer 3-planar graphs are quasiplanar,which implies the statement, by Theorem 1. Refer to Appendix A for details. (cid:117)(cid:116)
We first present a lower bound construction for this class of graphs:
Theorem 5.
There exist infinitely many -layer -planar graphs with n verticesand n − edges.Proof. We describe a family of graphs where p = q ; see Fig. 5a. Each topologicalgraph G consists of a sequence ( b , . . . , b β ) of K , -bricks such that b i and b i +1 share a planar edge for 1 ≤ i ≤ β −
1. Then G has n = 4 β + 2 vertices and m = 8 β + 1 = 2 n − (cid:117)(cid:116) Next, we provide a matching upper bound.
Theorem 6.
Any -layer -planar graph on n vertices has at most n − edges.Proof (Sketch). We first prove that in an optimal topological 2-layer 4-planargraph G , every triple of pairwise crossing edges is such that removing the tripleand at most four other edges separates G into two subgraphs G and G as shownin Fig. 5b. Based on this observation, we apply induction on the number of suchtriples in G . Note that in the base case, i.e., no triples of pairwise crossing edgesexist, the graph is quasiplanar. Refer to Appendix A for details. (cid:117)(cid:116) We first provide a lower bound construction for this class of graphs:
Theorem 7.
There exist infinitely many -layer -planar graphs with n verticesand n − edges. v u p v p · · ·· · ·· · · (a) u u u u v v v v (b) Fig. 6: (a) A family of 5-planar graphs on n = 2 p vertices with n − edges.(b) Graph S with n = 8 vertices and m = 14 > · − = 13 . u i v x u j u s v y G G u p v q u v u h v t (a) u p v q u = u i v = v x u j v t u s v y G u h u h (cid:48) v z v z (cid:48) (b) u p v q u h (cid:48) v z (cid:48) u j v y u j +1 v y +1 G (cid:48) (c) Fig. 7: (a) A triple ( u i , v y ), ( u s , v t ), ( u j , v x ) of pairwise crossing edges and at mostsix other edges separates an optimal 2-layer 5-planar graph into subgraphs G and G . If G consists of a single edge, (b) there can be edges ( u s , v z ), ( u s , v z (cid:48) ),( v t , u h ), ( v t , u h (cid:48) ), in which case (c) G consists of a graph G (cid:48) , vertices u j , v y andat most four of the green edges. Proof.
We augment the construction from Theorem 5 by a path of length β − β is the number of K , subgraphs; see the dashed blue edges in Fig. 6a.The obtained graph has n = 4 β + 2 vertices and m = 9 β = n − edges. (cid:117)(cid:116) For the specific value n = 8, we can provide a denser lower bound construction. Observation 5.
There exists a topological -layer -planar graph S with n = 8 vertices and m = 14 > n − edges; see Fig. 6b. We show that the graph S is in fact an exception, by demonstrating that thelower bound construction in Theorem 7 is tight for all other values of n . Theorem 8.
Any -layer -planar graph on n ≥ vertices has at most n − edges, except for graph S which has vertices and edges.Proof (Sketch). First observe that the theorem is clearly fulfilled if G = S .Otherwise, we apply an argument similar to the proof of Theorem 6. Namely,we first prove that if there is a triple of pairwise crossing edges in an optimaltopological 2-layer 5-planar graph, the removal of few edges separates the graphinto two components G and G ; see Fig. 7a. We then apply induction on thenumber of such triples in G . In particular, we consider some special cases, namely G could be S or a single edge; see also Fig. 7b. In the latter case, we alsoinvestigate the structure of graph G in more careful detail to prove our result;see also Fig. 7c. Refer to Appendix A for more details. (cid:117)(cid:116) A Crossing Lemma and General Density Bounds
In this section we generalize the well-known Crossing Lemma [6,28,35] to a metaCrossing Lemma for general graphs (Theorem 9), which also yields a densityupper bound for k -planar graphs. We denote by R a restriction on graphs, e.g., R can be “bipartite” or “2-layer”. We assume that for a fixed t >
0, there are α i , β i ∈ R for i ∈ { , . . . , t − } such that m ≤ α i n − β i is an upper boundfor the number of edges in R -restricted i -planar graphs. Let α := (cid:80) t − i =0 α i and β := (cid:80) t − i =0 β i . The proof of the next theorem follows the probabilistic techniqueof Chazelle, Sharir and Welzl (see e.g. [5, Chapter 35]); see also Appendix B. Theorem 9.
Let G be a simple R -restricted graph with n ≥ vertices and m ≥ α t n edges. The following inequality holds for the crossing number cr ( G ) : cr ( G ) ≥ t α m n . (1)The meta Crossing Lemma is used to obtain the following theorem regardingthe density. We follow closely the proof for corresponding statements for k -planarand bipartite k -planar graphs [2,9] in Appendix B. Theorem 10.
Let G be a simple R -restricted k -planar graph with n ≥ verticesfor some k ≥ t . Then m ≤ max (cid:40) , (cid:114) t √ k (cid:41) · α t n. We apply Theorems 9 and 10 to 2-layer k -planar graphs for t = 6. By [22],Theorems 2, 4, 6 and 8, we have ( α , α , α , α , α , α ) = (1 , , , , , ), yielding α = . By substituting the numbers in Theorem 9 we obtain the following. Corollary 1.
Let G be a simple -layer graph with n ≥ vertices and m ≥ n edges. Then, the following inequality holds for the crossing number cr ( G ) : cr ( G ) ≥ . . m n ≈ . m n . By plugging the result into Theorem 10 we obtain.
Corollary 2.
Let G be a simple -layer k -planar graph with n ≥ vertices forsome k > . Then m ≤ max (cid:26) , √ k (cid:27) · n. Note that for 2-layer 6-planar graphs, Corollary 2 certifies that m ≤ . n .We can show that there is only a gap of 0 . n towards an optimal solution: Theorem 11.
There exist infinitely many -layer -planar graphs with n verticesand n − edges. v u p v p · · ·· · ·· · · Fig. 8: A family of 6-planar graphs on n = 2 p vertices with n − v i · · ·· · · v i + r u i v i +1 v i + j · · · (a) v i · · ·· · · v i + r u i v i +1 v i − j · · · u i +1 u i − j + (cid:96) · · · (b) v i · · ·· · · v i + r u i v i +1 v i + j +1 u i + j · · · v i + j + (cid:96) · · · (c) v i · · ·· · · v i + r u i v i +1 u i +1 · · · u i − j · · · v i − j + (cid:96) (d) Fig. 9: Illustrations for the proof of Theorem 12.
Proof.
We augment the construction from Theorem 7 by a path of length β − β is the number of K , subgraphs; refer to the dotted blue path in Fig. 8.The obtained graph has n = 4 β + 2 vertices and m = 10 β − n − (cid:117)(cid:116) In the next theorem, we additionally show that the multiplicative constantfrom Corollary 2 is within a factor of 1 .
84 of the optimal achievable upper bound.
Theorem 12.
For any k , there exist infinitely many -layer k -planar graphswith n vertices and m = (cid:106)(cid:112) k/ (cid:107) n − O ( f ( k )) ≈ . √ kn − O ( f ( k )) edges.Proof (Sketch). We choose p = q and a parameter (cid:96) = (cid:98) (cid:112) k/ (cid:99) . We connectvertex u i to the (cid:96) vertices v i +1 . . . , v i + (cid:96) and vertex v i to vertices u i +1 . . . , u i + (cid:96) .Note that by symmetry, u i is also incident to the (cid:96) vertices v i − . . . , v i − (cid:96) andvertex v i to vertices u i − . . . , u i − (cid:96) . Clearly, this gives the density bound in thestatement of the theorem. Then, we consider an edge ( u i , v i + r ) and the crossingsit forms with edges incident to some other vertices; see Fig. 9. This allows us toestablish that each edge has at most k crossings. For details, see Appendix B. (cid:117)(cid:116) k -Planar Graphs In this section, we present some properties of 2-layer k -planar graphs.In Theorem 4, we have established that every optimal 2-layer 3-planar graphis (3-)quasiplanar, which is also the case in the general, non-layered, drawingmodel [14]. A more general relationship between the classes of k -planar and (cid:48) u (cid:48) h − u (cid:48) u (cid:48) h v (cid:48) v (cid:48) v (cid:48) h v (cid:48) h − · · ·· · · Fig. 10: A set of h pairwise crossing edges in a topological 2-layer graph. h -quasiplanar graphs was uncovered in [7], where it is proven that every k -planargraph is ( k + 1)-quasiplanar, for every k ≥
2. Next, we show that for 2-layerdrawings an even stronger relationship holds.
Theorem 13.
For k ≥ , every -layer k -planar graph is -layer (cid:6) k + 2 (cid:7) -quasiplanar. Further, every -layer -planar graph is -layer (3-)quasiplanar.Proof. Let G be a topological 2-layer k -planar graph, with k ≥
3, which we assumew.l.o.g. to be connected. Suppose for a contradiction that G contains h := (cid:100) k +2 (cid:101) mutually crossing edges ( u (cid:48) i , v (cid:48) h +1 − i ) for 1 ≤ i ≤ h in G , such that u (cid:48) , . . . , u (cid:48) h and v (cid:48) , . . . , v (cid:48) h appear in this order in u , . . . , u p and v , . . . , v q , respectively. Observethat ( u (cid:48) , v (cid:48) h ) and ( v (cid:48) , u (cid:48) h ) have h − h -tuple. Moreover, bothendvertices of all the h − u (cid:48) i , v (cid:48) h +1 − i ), for i = 2 , . . . , h −
1, are located inregions bounded by e (1) := ( u (cid:48) , v (cid:48) h ) and e (2) := ( v (cid:48) , u (cid:48) h ); see Fig. 10. Since G isconnected, for each 2 ≤ i ≤ h −
1, the edge ( u (cid:48) i , v (cid:48) h +1 − i ) is adjacent to anotheredge e i . Note that either e i = e j for some j (cid:54) = i , and e i crosses e (1) and e (2) ,or e i (cid:54) = e j for all j (cid:54) = i , and e i crosses one of e (1) and e (2) . This implies h − { e (1) , e (2) } , and, consequently, e (1) or e (2) is crossed byat least h − (cid:100) ( h − / (cid:101) edges. We obtain h − (cid:100) ( h − / (cid:101) ≥ h − ≥ (cid:0) k + 2 (cid:1) − k + 1 crossings for e (1) or e (2) , a contradiction.For the case k = 2, assume that G contains three mutually crossing edges e = ( u (cid:48) , v (cid:48) ), e = ( u (cid:48) , v (cid:48) ) and e = ( u (cid:48) , v (cid:48) ), such that u (cid:48) , u (cid:48) , u (cid:48) and v (cid:48) , v (cid:48) , v (cid:48) .appear in this order in u , . . . , u p and v , . . . , v q , respectively. As e and e arealready crossed twice, e represents a connected component; contradiction. (cid:117)(cid:116) Next, we show that the pathwidth of 2-layer k -planar graphs is bounded by k + 1. We point out that similar results are known for layered graphs with abounded total number of crossings [26] and for layered fan-planar graphs [15],and that these bounds do not have any implication on 2-layer k -planar graphs. Theorem 14.
Every -layer k -planar graph has pathwidth at most k + 1 .Proof. Let G be a topological 2-layer k -planar graph with parts U and V . We firstdefine a total ordering ≺ on the edges as follows: We say that edge e = ( u i , v x )precedes edge e = ( u j , v y ), or e ≺ e , if u i , u j ∈ U and either (i) i < j , or(ii) i = j and x < y . Let E = ( e , . . . , e m ) be the set of edges ordered with respectto ≺ . Let e i = ( u s , v t ) be an edge and let v y be a vertex in V . Further let e y − and e y + be the first and the last edge incident to v y in ≺ , respectively. We call y (cid:48) u y (cid:48)(cid:48) v y u i (a) u y (cid:48) u y (cid:48)(cid:48) v y u i (b) u y (cid:48) = u i u y (cid:48)(cid:48) v y v z (c) u y (cid:48)(cid:48) = u i u y (cid:48) v y v z (d) Fig. 11: Illustrations for the proof of Theorem 14. v y related to e i if v y is incident to an edge crossing e i and if y − < i < y + . Forevery edge e i = ( u s , v t ) ∈ E , we construct a bag B i that contains u s , v t and allthe (at most k ) related vertices of e i . Then, we connect B i to bags B i − and B i +1 (if they exist), obtaining a path of bags P .In the following we show that P is a valid path decomposition of G . Since weassigned at most k + 2 vertices to each bag of P the width of P is at most k + 1.Properties P.1 and P.3 of a tree decomposition are fulfilled for P by construction.We may assume that G is connected, otherwise we compute a path decompositionfor each connected component and link the obtained vertex disjoint paths. Hencealso P.2 is fulfilled. Moreover, by the choice of ≺ , all the edges incident to a vertex u i ∈ U occur in a consecutive sequence, i.e. u i is incident to edges e j , . . . , e k forsome 1 ≤ j ≤ k ≤ m and then u i appears in all of bags B j , . . . , B k , which is asubpath of P . Therefore, Property P.4 also holds for all vertices in U .It remains to show that Property P.4 holds for every vertex v y ∈ V . Let e y − =( u y (cid:48) , v y ) and e y + = ( u y (cid:48)(cid:48) , v y ). Note that each of the edges e y − , e y − +1 , . . . , e y + iseither incident to v y (see Fig. 11a), or it crosses one of e y − and e y + , since itsendvertex in U is some u i with y (cid:48) ≤ i ≤ y (cid:48)(cid:48) ; see Figs. 11b to 11d. Note thatfor the endvertex v z in V necessarily z > y if u i = u y (cid:48) or z < y if u i = u y (cid:48)(cid:48) bydefinition of ≺ ; see Fig. 11c or Fig. 11d, respectively. Hence v y belongs to allbags B y − , B y − +1 , . . . , B y + and P.4 holds. The statement follows. (cid:117)(cid:116) We gave results for 2-layer k -planar graphs regarding their density, relationshipto 2-layer h -quasiplanar graphs, and pathwidth. Tight density bounds for 2-layer k -planar graphs with k = 6 may be achievable following similar arguments to theproof of Theorem 8, which would also improve upon our results for the CrossingLemma, and in turn on the density for general k . Moreover, a better lower boundfor general k may exist. The relationship to other beyond-planar graph classes isalso of interest. With respect to the pathwidth, we conjecture that our upperbound is tight. Finally, the recognition and characterization of 2-layer k -planargraphs remain important open problems. eferences
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We start with the full proof of Lemma 1.
Lemma 1.
For k ≥ , there exist positive rational numbers a k ≥ and b k ≥ such that (i) every n -vertex -layer k -planar graph has at most a k n − b k edgesfor n ≥ n k with n k a constant, and (ii) there is a -layer k -planar graph with n vertices and exactly a k n − b k edges for some n > .Proof. First observe that the number of edges in an n -vertex k -planar graph canbe upper bounded by a linear function g ( n ) [2,39]. Thus in an n -vertex 2-layer k -planar graph, there can be at most f ( n ) ≤ g ( n ) edges for a linear function f ( n ). If the multiplicative factor a k in f ( n ) = a k n − o ( n ) is irrational, one canchoose a slightly larger rational multiplicative factor a (cid:48) k . So assume w.l.o.g. that a k is rational. Since optimal 2-layer 1-planar graphs can have n − a k ≥ .Because a k is rational and the number of edges in a graph must be integer,there is a limited number of possible rational differences between the maximumdensity and the function a k n . We choose the smallest divergence as our additiveconstant b k . (cid:117)(cid:116) The following lemma is a key ingredient in the proof of the density upperbound for 2-layer 3-planar graphs in Theorem 4.
Lemma 4.
Let ( u i , v y ) , ( u s , v t ) and ( u j , v x ) be a triple of pairwise crossingedges in a topological -layer -planar graph such that ≤ i < s < j ≤ p and ≤ x < t < y ≤ q . Then the number of edges incident to u s or v t is at most .Proof. Consider such a triple of edges. If u s is connected to another vertex v (cid:54) = v t ,edge ( u s , v ) crosses one of ( u i , v y ) and ( u j , v x ). The same is true, if v t is connectedto another vertex u (cid:54) = u s . Since ( u i , v y ) and ( u j , v x ) both have two crossingsfrom the triple of crossing edges, u s and v t can only be incident to a total of twoedges which are not ( u s , v t ). (cid:117)(cid:116) Theorem 4.
Let G be a topological -layer -planar graph on n vertices. Then G has at most n − edges for n ≥ . Moreover, if G is optimal, it is quasiplanar.Proof. Let G be an optimal 2-layer 3-planar graph on n vertices. By Lemma 4,for every triple of pairwise crossing edges there exists two vertices u s and v t which are incident to a total of at most 4 edges. Removing u s and v t reducesthe number of edges by at most 3 and the number of vertices by 2. Since byTheorem 3 optimal 2-layer 3-planar graphs have density at least 2 n −
4, theremoval of u s and v t yields a denser subgraph. We conclude that G contains notriple of pairwise crossing edges. Thus, G is quasiplanar and has at most 2 n − n ≥
2, by Theorem 1. (cid:117)(cid:116)
Theorem 6.
Any -layer -planar graph on n vertices has at most n − edges.roof. Consider an optimal 2-layer 4-planar graph G with exactly a n − b edges.By Theorems 1 and 5, this graph cannot be quasiplanar and hence contains a tripleof pairwise crossing edges ( u i , v y ), ( u s , v t ) and ( u j , v x ) for some 1 ≤ i < s < j ≤ p and 1 ≤ x < t < y ≤ q . We first show that there is no vertex u s (cid:48) such that i < s (cid:48) < j and s (cid:48) (cid:54) = s . Assume for a contradiction that such a vertex u s (cid:48) exists.Each of the edges ( u i , v y ) and ( u j , v x ) have two crossings from the triple ofpairwise crossing edges, so each of them can only be crossed by two more edges.Hence, there are at most five edges incident to the vertices u s , u s (cid:48) , v t (includingthe edge ( u s , v t )). Then the graph G (cid:48) obtained by removing vertices u s , u s (cid:48) , v t has n (cid:48) = n − m (cid:48) ≥ m − a ( n − − b + (3 a − > a n (cid:48) − b edges; a contradiction to the optimality of G . Symmetrically, there is no vertex v t (cid:48) such that x < t (cid:48) < y and t (cid:48) (cid:54) = t . So we have s = i + 1, j = i + 2, t = x + 1and y = x + 2.Next, consider the subgraph G induced by u , . . . , u i and v , . . . v x andthe subgraph G induced by u i +2 , . . . , u p and v x +2 , . . . v q . We show that G isconnected to G only by the edges ( u i , v x +2 ), ( u i +2 , v x ) and paths traversing u s or v t ; see Fig. 5b. Assume for a contradiction that there is an edge ( u h , v z ) suchthat w.l.o.g. 1 ≤ h ≤ i and x + 2 ≤ z ≤ q . This edge would cross ( u s , v t ) andat least one of the edges ( u i , v x +2 ) and ( u i +2 , v x ), say ( u i +2 , v x ). Then, ( u h , v z ),( u i +2 , v x ), and ( u s , v t ) would form a triple of pairwise crossing edges where u i and u i +1 are between u h and u i +2 ; a contradiction to the previously establishedclaim.We show by induction on the number of triples of pairwise crossing edges thatthe number of edges of G is at most 2 n −
3. For the base case, assume that thereis no triple of pairwise crossing edges. Then, G is quasiplanar and has at most2 n − n = 2, it has at most 1 = 2 n − G has a triple of pairwise crossing edgesthat connects subgraphs G and G as described above. Clearly, G and G have less triples of pairwise crossing edges than G . As mentioned before, since( u i , v x +2 ) and ( u i +2 , v x ) have two crossings from the triple of pairwise crossingedges, u i +1 and v x +1 can only be incident to a total of five edges including edge( u i +1 , v x +1 ). Hence, G can be split into (possibly optimal) subgraphs G and G and two isolated vertices u i +1 and v x +1 by removing seven edges. Let n and n be the number of vertices of G and G , respectively. Clearly, n = n + n + 2.By induction, G and G have at most 2 n − n − m ≤ n + n ) − n + n ) + 1 = 2( n −
2) + 1 = 2 n − (cid:117)(cid:116) Theorem 8.
Any -layer -planar graph on n ≥ vertices has at most n − edges, except for graph S which has vertices and edges.Proof. Consider an optimal topological 2-layer 5-planar graph G . If G = S , thetheorem trivially holds. Hence, assume that G (cid:54) = S has exactly a n − b edges.By Theorems 1 and 7, it cannot be quasiplanar and hence contains a triple ofpairwise crossing edges ( u i , v y ), ( u s , v t ) and ( u j , v x ) for some 1 ≤ i < s < j ≤ p and 1 ≤ x < t < y ≤ q . We first show that there is at most one vertex u s (cid:48) suchhat i < s (cid:48) < j and s (cid:48) (cid:54) = s , or at most one vertex v t (cid:48) such that x < t (cid:48) < y and t (cid:48) (cid:54) = t . Assume for a contradiction that two such vertices w and w (cid:48) exist. Sinceboth edges ( u i , v y ) and ( u j , v x ) have two crossings from the triple of pairwisecrossing edges, each of those edges can only be crossed by three more edges.Hence, there are at most seven edges incident to vertices u s , v t , w, w (cid:48) (includingthe edge ( u s , v t )). By removing the four vertices u s , v t , w, w (cid:48) together with the atmost seven incident edges, we would obtain a graph G (cid:48) with n (cid:48) = n − m (cid:48) = m − a n − b − a ( n − − b + (4 a −
7) edges. Given a ≥ by Theorem 7, it holds that 4 a − ≥ m (cid:48) > a n (cid:48) − b ; a contradiction.Next, consider the subgraph G induced by vertices u , . . . , u i and v , . . . v x and the subgraph G induced by vertices u j , . . . , u p and v y , . . . v q . We show that G is connected to G only by the edges ( u i , v y ), ( u j , v x ), some paths traversing u s or v t and potentially an edge ( u h , v z ) such that h ∈ { i, j } or z ∈ { x, y } ; seeFig. 7a. Assume for a contradiction that there is an edge ( u h , v z ) such thatw.l.o.g. 1 ≤ h < i and y < z ≤ q . This edge would cross all three edges ( u i , v y ),( u j , v x ), and ( u s , v t ). Then, ( u h , v z ), ( u j , v x ), and ( u s , v t ) form a triple of pairwisecrossing edges where u i and v y are between u h and u j , and between v x and v z ,respectively; a contradiction to the previously established claim. By a similarargument, there can be only one such edge ( u h , v z ). In the following, we assumew.l.o.g. that h < i and y = z .If both, ( u i , v y ), ( u j , v x ), and ( u s , v t ) as well as ( u h , v y ), ( u j , v x ), and ( u s , v t ),form triples of pairwise crossing edges for 1 ≤ h < i < s < j ≤ p and 1 ≤ x < t < y ≤ q , we call the triple ( u h , v y ), ( u j , v x ) and ( u s , v t ) maximal . Notethat in this maximal triple, vertex u i is between u h and u j . On the other hand,if ( u i , v y ), ( u j , v x ), and ( u s , v t ) form a triple of pairwise crossing edges with1 ≤ i < s < j ≤ p and 1 ≤ x < t < y ≤ q such that there is no h so that ( u h , v y ),( u j , v x ), and ( u s , v t ) also forms a triple of pairwise crossing edges with 1 ≤ h < i ,we call the triple ( u i , v y ), ( u j , v x ), and ( u s , v t ) maximal. We show by induction onthe number of maximal triples of pairwise crossing edges 1 ≤ h < i < s < j ≤ p and 1 ≤ x < t < y ≤ q that the number of edges of G (cid:54) = S is at most n − for n ≥
3, while the number of edges of G is at most 1 for n = 2. For the basecase, assume that there is no triple of pairwise crossing edges or that G = S .In the former case, G is quasiplanar and has at most 2 n − n − for n ≥
3, while it clearly can only have one edge if n = 2. In the latter case, i. e. G = S , graph G has 14 edges.For the induction step, we will assume that G has a triple of maximal pairwisecrossing edges that connects subgraphs G and G as described above. Clearly, G and G have fewer maximal triples of pairwise crossing edges than G . Asmentioned before, since ( u i , v y ) and ( u j , v x ) have two crossings from the tripleof pairwise crossing edges, all vertices between u i and u j (which are at mostthree), and v x and v y , respectively, are incident to a total of at most sevenedges including the edge ( u s , v t ). Hence, G can be split into (possibly optimal)subgraphs G and G , two isolated vertices u s , v t , and possibly one more vertex u s (cid:48) between u i and u j by removing nine edges. Note that if u s (cid:48) exists, since everyincidence to u s , v t and u s (cid:48) implies a crossing on ( u i , v y ) or ( u j , v x ), one of u s , v t nd u s (cid:48) would have degree at most two. Let w denote this vertex. Then the graph G (cid:48) obtained by removing w has n (cid:48) = n − m (cid:48) = m − ≤ n (cid:48) − edges since G (cid:48) (cid:54) = S . Then, m ≤ ( n − − + 2 = n − . .Let n and n denote the number of vertices of G and G , respectively.Clearly, n ≥ n + n + 2. Assume first that w.l.o.g. G is isomorphic to S . Weobserve that ( u , v ) is a planar edge of G , since edges ( u , v ) and ( v , u ) havefive crossings each within S ; see Fig. 6b. Then, consider the graph G (cid:48) obtainedfrom G by the removal of vertices u , u , u , v , v and v . Here we considertwo cases. If G (cid:48) is also isomorphic to S , then G contains n = 14 vertices and m = 27 = · − edges. Otherwise, G (cid:48) has n (cid:48) = n − m (cid:48) ≤ n (cid:48) − edges. Then, m = m (cid:48) + 13 ≤ ( n − − + 13 = n − .Next, assume that n , n ≥
3. Since we already covered the case where G = S ,we may assume that G and G are not isomorphic to S . Then, by induction, G and G have at most n − and n − edges, respectively. We conclude that m ≤ ( n + n ) − · + 9 ≤ ( n + n ) − ( n + n ) ≤ ( n −
2) = n − .Finally, consider the case where n = 2; see Fig. 7b. Because u i belongs to G , u i can only be incident to v x , v t and v y . Hence, edge ( u i , v t ) must existsince otherwise u i would have degree two. Symmetrically, edge ( u s , v x ) is alsopresent. Then, u s and v t can only be incident to a total of 7 edges, if thereare edges ( u s , v z ), ( u s , v z (cid:48) ), ( u h , v t ) and ( u h (cid:48) , v t ) for some j ≤ h < h (cid:48) ≤ p and y ≤ z < z (cid:48) ≤ q . If u s and v t were only incident to at most 6 edges, G has m = m − G or u s or u t and n = n − m ≤ n − , wecan conclude that G has at most m = m + 9 ≤ ( n −
4) + = n − edges.Therefore, we assume in the following that n ≥ n = 4, we observe that G has 8 vertices and hence has less than n − edges, or exactly 14 edges if it is S . Thus, assume that, n >
4. We observe thatall edges in G that are incident to u j will cross edge ( u h (cid:48) , v t ). Since ( u h (cid:48) , v t )already has three crossings, it follows that the degree of u j in G is at mosttwo. Symmetrically, the degree of v y in G is at most two. Consider the graph G (cid:48) obtained from G by removing u j and v y ; see Fig. 7c. Since n > G (cid:48) has n (cid:48) ≥ G (cid:48) is isomorphic to S . Then, edge ( u j +1 , v y +1 )is planar, and u j and v y can only be incident to ( u j , v y ), ( u j , v y +1 ) and ( u j +1 , v y ).Then, G has n = 10 vertices and m = 17 = n − edges.Next, assume that G (cid:48) is not isomorphic to S . We consider two cases. If ( u j , v y )is not in G consider the graph G ∗ obtained from G by inserting edge ( u j , v y ).Clearly, G ∗ is 2-layer 5-planar and hence has at most m ∗ ≤ n − edges. Since m = m ∗ −
1, it follows that m ≤ n − . So assume that ( u j , v y ) is part of G .Then, there are only three edges in G that are not in G (cid:48) . Assume that G (cid:48) has m (cid:48) ≤ n (cid:48) − edges. Then, G has m = m (cid:48) + 3 ≤ ( n − − + 3 = n − edges.We conclude that m ≤ n − . Thus, m ≤ m + m +9 = 1+ n − +9 ≤ ( n − − + 10 = ( n −
4) + = n − = n − . (cid:117)(cid:116) Omitted Proofs from Section 4
We first give an auxiliary lemma that will be used in the proof of Theorem 9.
Lemma 5.
Let G be a simple R -restricted graph with n ≥ vertices and m edges. Then, the following inequality holds for the crossing number cr ( G ) : cr ( G ) ≥ tm − αn + β. (2) Proof.
Clearly, Inequality (2) holds for m ≤ α n − β . Next, assume that m >α n − β . Then there exists at least m − ( α n − β ) edges in G that have at leastone crossing. If m > α n − β , there exists at least m − ( α n − β ) edges in G that have at least two crossings. Iteratively we obtain that, if m > α i − n − β i − ,there exists at least m − ( α i n − β i ) edges in G that have at least i crossings.Therefore we obtain cr ( G ) ≥ t − (cid:88) i =0 [ m − ( α i n − β i )] = tm − αn + β which concludes the proof. (cid:117)(cid:116) Theorem 9.
Let G be a simple R -restricted graph with n ≥ vertices and m ≥ α t n edges. The following inequality holds for the crossing number cr ( G ) : cr ( G ) ≥ t α m n . (1) Proof.
Consider a drawing Γ of G with cr ( G ) crossings and let π = αn tm ≤ π choose every vertex of G independently and let G π denotethe subgraph of G induced by the chosen vertices, and Γ π the subdrawing of Γ representing G π . Consider random variables N π , M π , and C π denoting thenumber of vertices and edges in G π and the number of crossings in Γ π , respectively.By Lemma 5, it holds that C π ≥ tM π − αN π + β . Taking expectations on thisrelationship, we obtain: π cr ( G ) ≥ tπ m − απn = ⇒ cr ( G ) ≥ tmπ − αnπ We obtain Inequality (1) by substituting π = αn tm into the inequality above. (cid:117)(cid:116) Theorem 10.
Let G be a simple R -restricted k -planar graph with n ≥ verticesfor some k ≥ t . Then m ≤ max (cid:40) , (cid:114) t √ k (cid:41) · α t n. Proof. If m ≤ α t n , the proof follows immediately. Otherwise, we obtain fromTheorem 9 and from the assumption that G is k -planar:4 t α m n ≤ cr ( G ) ≤ mk. his implies: m ≤ α t (cid:114) t √ kn which completes the proof. (cid:117)(cid:116) Theorem 12.
For any k , there exist infinitely many -layer k -planar graphswith n vertices and m = (cid:106)(cid:112) k/ (cid:107) n − O ( f ( k )) ≈ . √ kn − O ( f ( k )) edges.Proof. We choose p = q . Depending on k , we choose a parameter (cid:96) that we willcalculate later. We connect vertex u i to the (cid:96) vertices v i +1 . . . , v i + (cid:96) . Similarly,we connect vertex v i to vertices u i +1 . . . , u i + (cid:96) . Orient the edges from lower tohigher index. Then, each vertex (except for those with indices at least p − (cid:96) ) has (cid:96) outgoing edges. Moreover, edge ( u i , v i + r ) (for 1 ≤ r ≤ (cid:96) ) is only crossed by– all (cid:96) outgoing edges from vertex v i + j for 0 ≤ j ≤ r −
1; see Fig. 9a,– (cid:96) − j outgoing edges from vertex v i − j for 1 ≤ j ≤ (cid:96) − v i − j tovertices u i +1 , . . . , u i + (cid:96) − j ; see Fig. 9b,– by r − − j outgoing edges from vertex u i + j for 1 ≤ j ≤ r − u i + j to vertices v i + j +1 , . . . , v r − ; see Fig. 9c, and,– by (cid:96) − r − j − u i − j for 1 ≤ j ≤ (cid:96) − r − u i − j to vertices v i + r +1 , . . . , v i + (cid:96) − j ; see Fig. 9d.In total, for the number of edges crossing ( u i , v i + r ) we have r(cid:96) + (cid:96) − (cid:88) i =1 i + r − (cid:88) i =1 i + (cid:96) − r − (cid:88) i =1 i = (cid:96)r + ( (cid:96) − (cid:96) )2 + ( r − r − (cid:96) − r − (cid:96) − r − ≤ (cid:96)r + (cid:96) r (cid:96) − r ) (cid:96) + r . The last term is maximal for r = (cid:96) , yielding at most 2 (cid:96) crossings on ( u i , v i + (cid:96) ). Toensure k -planarity we set 2 (cid:96) ≤ k and obtain (cid:96) ≤ (cid:112) k/
2, which implies that everyvertex (except for those with the (cid:96) = O ( f ( k )) largest indices) has (cid:96) = (cid:112) k/2outgoing edges. The statement follows.