Crossing numbers and combinatorial characterization of monotone drawings of K n
aa r X i v : . [ m a t h . C O ] A ug Crossing numbers and combinatorial characterization ofmonotone drawings of K n ∗ Martin Balko † Radoslav Fulek ‡ Jan Kynˇcl § Abstract
In 1958, Hill conjectured that the minimum number of crossings in a drawing of K n is exactly Z ( n ) = ⌊ n ⌋ (cid:4) n − (cid:5) (cid:4) n − (cid:5) (cid:4) n − (cid:5) . Generalizing the result by ´Abrego et al. for2-page book drawings, we prove this conjecture for plane drawings in which edges arerepresented by x -monotone curves. In fact, our proof shows that the conjecture remainstrue for x -monotone drawings of K n in which adjacent edges may cross an even number oftimes, and instead of the crossing number we count the pairs of edges which cross an oddnumber of times. We further discuss a generalization of this result to shellable drawings,a notion introduced by ´Abrego et al. We also give a combinatorial characterization ofseveral classes of x -monotone drawings of complete graphs using a small set of forbiddenconfigurations. For a similar local characterization of shellable drawings, we generalizeCarath´eodory’s theorem to simple drawings of complete graphs. Let G be a graph with no loops or multiple edges. In a drawing D of a graph G in theplane, the vertices are represented by distinct points and each edge is represented by a simplecontinuous arc connecting the images of its endpoints. As usual, we identify the vertices andtheir images, as well as the edges and the arcs representing them. We require that the edgespass through no vertices other than their endpoints. We also assume for simplicity that anytwo edges have only finitely many points in common, no two edges touch at an interior pointand no three edges meet at a common interior point.A crossing in D is a common interior point of two edges where they properly cross. The crossing number cr( D ) of a drawing D is the number of crossings in D . The crossing number ∗ The authors were supported by the grant GA ˇCR GIG/11/E023 GraDR in the framework of ESF EU-ROGIGA program. The first and the third author were also supported by the Grant Agency of the CharlesUniversity, GAUK 1262213, and by the grant SVV-2013-267313 (Discrete Models and Algorithms). The thirdauthor was also partially supported by ERC Advanced Research Grant no 267165 (DISCONV). The secondauthor gratefully acknowledges support from the Swiss National Science Foundation Grant PBELP2 146705. † Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Mal-ostransk´e n´am. 25, 118 00 Praha 1, Czech Republic; [email protected] ‡ Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Mal-ostransk´e n´am. 25, 118 00 Praha 1, Czech Republic; and IEOR, Columbia University, NYC, NY, USA; [email protected] § Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University,Faculty of Mathematics and Physics, Malostransk´e n´am. 25, 118 00 Praha 1, Czech Republic; and Alfr´edR´enyi Institute of Mathematics, Re´altanoda u. 13-15, Budapest 1053, Hungary; [email protected] G ) of a graph G is the minimum of cr( D ), taken over all drawings D of G . A drawing D is called simple if no two adjacent edges cross and no two edges have more than one commoncrossing. It is well known and easy to see that every drawing of G which minimizes thecrossing number is simple.According to the famous conjecture of Hill [21, 23] (also known as Guy’s conjecture), thecrossing number of the complete graph K n on n vertices satisfies cr( K n ) = Z ( n ), where Z ( n ) = 14 (cid:22) n (cid:23) (cid:22) n − (cid:23) (cid:22) n − (cid:23) (cid:22) n − (cid:23) . This conjecture has been verified for n ≤
10 by Guy [22] and recently for n ≤
12 by Pan andRichter [31]. Moreover for each n , there are drawings of K n with exactly Z ( n ) crossings [11,21, 23, 24]. Current best asymptotic lower bound, cr( K n ) ≥ . Z ( n ), follows from thelower bound on the crossing number of the complete bipartite graph [25] by an elementarydouble-counting argument [35].A curve α in the plane is x-monotone if every vertical line intersects α in at most onepoint. A drawing of a graph G in which every edge is represented by an x -monotone curve andno two vertices share the same x -coordinate is called x -monotone (or monotone , for short).The monotone crossing number mon-cr( G ) of a graph G is the minimum of cr( D ), taken overall monotone drawings D of G .The rectilinear crossing number cr( G ) of a graph G is the smallest number of crossingsin a drawing of G where every edge is represented by a straight-line segment. Since everyrectilinear drawing of G in which no two vertices share the same x -coordinate is x -monotone,we have cr( G ) ≤ mon-cr( G ) ≤ cr( G ) for every graph G .The odd crossing number ocr( G ) of a graph G is the minimum number of pairs of edgescrossing an odd number of times in a drawing of G in the plane. The monotone odd crossingnumber , mon-ocr( G ), is the minimum number of pairs of edges crossing an odd number oftimes in a monotone drawing of G . For these two notions of the crossing number, optimaldrawings do not have to be simple. Moreover, there are graphs G with ocr( G ) < cr( G ) [32, 41],and for every n , there is a graph G with mon-ocr( G ) = 1 and mon-cr( G ) ≥ n [17].We call a drawing of a graph semisimple if adjacent edges do not cross but independentedges may cross more than once. The monotone semisimple odd crossing number of G (called monotone odd + by Schaefer [38]), denoted by mon-ocr + ( G ), is the smallest number of pairsof edges that cross an odd number of times in a monotone semisimple drawing of G . Wecall a drawing of a graph weakly semisimple if every pair of adjacent edges cross an evennumber of times; independent edges may cross arbitrarily. The monotone weakly semisimpleodd crossing number of G , denoted by mon-ocr ± ( G ), is the smallest number of pairs of edgesthat cross an odd number of times in a monotone weakly semisimple drawing of G . Clearly,mon-ocr( G ) ≤ mon-ocr ± ( G ) ≤ mon-ocr + ( G ) ≤ mon-cr( G ) . The monotone crossing number has been introduced by Valtr [42] and recently furtherinvestigated by Pach and T´oth [30], who showed that mon-cr( G ) < G ) holds for everygraph G . On the other hand, they showed that the monotone crossing number and thecrossing number are not always the same: there are graphs G with arbitrarily large crossingnumbers such that mon-cr( G ) ≥ cr( G ) − . We study the monotone crossing numbers of complete graphs. The drawings of completegraphs with Z ( n ) crossings obtained by Blaˇzek and Koman [11] (see also [24]) are 2 -page book drawings. In such drawings the vertices are placed on a line l and each edge is fully contained2 p p p p p p p Figure 1: An example of a 2-page book drawing of K with Z (8) = 18 crossings obtained byBlaˇzek and Koman [11].in one of the half-planes determined by l . Since 2-page drawings may be considered as a strictsubset of x -monotone drawings, we have mon-cr( K n ) ≤ Z ( n ).´Abrego et al. [1] recently proved that Hill’s conjecture holds for 2-page book drawings ofcomplete graphs. We generalize their techniques and show that Hill’s conjecture holds for all x -monotone drawings of complete graphs, and even for the monotone weakly semisimple oddcrossing number. Theorem 1.1.
For every n ∈ N , we have mon-ocr ± ( K n ) = mon-ocr + ( K n ) = mon-cr( K n ) = Z ( n ) . The rectilinear crossing number of K n is known to be asymptotically larger than Z ( n ):this follows from the best current lower bound cr( K n ) ≥ (277 / (cid:0) n (cid:1) − O ( n ) [5, 7] and fromthe simple upper bound Z ( n ) ≤ (cid:0) n (cid:1) + O ( n ).See a recent survey by Schaefer [38] for an encyclopedic treatment of all known variantsof crossing numbers.During the preparation of this paper, we were informed that the authors of [1] achieved theresult mon-cr( K n ) = Z ( n ) already during discussions after their presentation at SoCG 2012and that Silvia Fernandez-Merchant was going to present it in her keynote talk at LAGOS2013. The proceedings of the conference were recently published [2]. Pedro Ramos [34]then presented the results and some further developments at the XV Spanish Meeting onComputational Geometry (ECG 2013) in his invited talk. Very recently, ´Abrego et al. [3]made their paper containing a more general result publicly available.In Section 2, we first prove Theorem 1.1 for semisimple monotone drawings. Then weextend the result to weakly semisimple monotone drawings, by showing that even crossingsof adjacent edges can be easily eliminated in such drawings.In Section 3 we introduce a combinatorial characterization of x -monotone drawings of K n .We show that there is a one-to-one correspondence between semisimple, simple or pseudolinear x -monotone drawings of K n and mappings (cid:0) [ n ]3 (cid:1) → { + , −} , called signature functions , avoidinga finite number of certain sub-configurations. The signature functions were introduced byPeters and Szekeres [40] as a generalization of order types of planar points sets.In Section 4 we show a further generalization of Theorem 1.1 to shellable drawings andweakly shellable drawings; we define these notions in the beginning of Section 4. We show a3ocal characterization of shellable drawings, for which we generalize Caratheodory’s theoremto simple drawings of complete graphs. We also show that shellable drawings form a moregeneral class than monotone drawings. Finally, we further generalize a key lemma from [1],which implies a generalization of the main result of [3] to weakly semisimple drawings.In the last section we state our stronger version of Hill’s conjecture. Let P denote a set of n points in the plane in general position and let k be an integer satisfying0 ≤ k ≤ n . The line segment joining a pair of points p and q in P is a k -edge ( ≤ k -edge ) ifthere are exactly (at most, respectively) k points of P in one of the open half-planes definedby the line pq .´Abrego and Fern´andez-Merchant [6] and Lov´asz et al. [28] discovered a relation betweenthe numbers of k -edges (or ≤ k -edges ) in P and the number of convex 4-tuples of points in P , which is equal to the number of crossings of the complete geometric graph with vertexset P . This relation transforms every lower bound on the number of ≤ k -edges to a lowerbound on the number of crossings. Using this method, many incremental improvements onthe rectilinear and pseudolinear crossing number of K n have been achieved [4, 5, 6, 8, 10, 28].To prove the lower bound on the 2-page crossing number of K n , ´Abrego et al. [1] gen-eralized the notion of k -edges to arbitrary simple drawings of complete graphs. They alsointroduced the notion of ≤≤ k -edges, which capture the essential properties of 2-page bookdrawings better than ≤ k -edges. We show that the approach using ≤≤ k -edges can be gener-alized to arbitrary semisimple x -monotone drawings.For a semisimple drawing D of K n and distinct vertices u and v of K n , let γ be the orientedarc representing the edge { u, v } . If w is a vertex of K n different from u and v , then we saythat w is on the left (right) side of γ if the topological triangle uvw with vertices u , v and w traced in this order is oriented counter-clockwise (clockwise, respectively). This generalizesthe definition introduced by ´Abrego et al. [1] for simple drawings. Further generalization ispossible for weakly semisimple drawings, where every two edges of the triangle uvw crossan even number of times; see Section 4. However, we were not able to find a meaningfulgeneralization of this notion to arbitrary drawings, where the edges of the triangle uvw cancross an odd number of times.A k -edge in D is an edge { u, v } of D that has exactly k vertices on the same side (left orright). Since every k -edge has n − − k vertices on the other side, every k -edge is also an( n − − k )-edge and so every edge of D is a k -edge for some integer k where 0 ≤ k ≤ ⌊ n/ ⌋− i -edge in D with i ≤ k is called a ≤ k -edge .Let E i ( D ) be the number of i -edges and E ≤ k ( D ) the number of ≤ k -edges of D . Clearly, E ≤ k ( D ) = P ki =0 E i ( D ). Similarly, the number E ≤≤ k ( D ) of ≤≤ k -edges of D is defined by thefollowing identity. E ≤≤ k ( D ) = k X j =0 E ≤ j ( D ) = k X i =0 ( k + 1 − i ) E i ( D ) . (1)Considering the only three different simple drawings of K up to a homeomorphism of theplane, ´Abrego et al. [1] showed that the number of crossings in a simple drawing D of K n canbe expressed in terms of the number of k -edges in the following way.4 emma 2.1 ([1]) . For every simple drawing D of K n we have cr( D ) = 3 (cid:18) n (cid:19) − ⌊ n/ ⌋− X k =0 k ( n − − k ) E k ( D ) , (2) which can be equivalently rewritten as cr( D ) = 2 ⌊ n/ ⌋− X k =0 E ≤≤ k ( D ) − (cid:18) n (cid:19) (cid:22) n − (cid:23) −
12 (1 + ( − n ) E ≤≤⌊ n/ ⌋− ( D ) . Lemma 2.1 generalizes the relation found by ´Abrego and Fern´andez-Merchant [6]. Wefurther generalize it to semisimple drawings of K n where cr( D ) is replaced by ocr( D ), whichcounts the number of pairs of edges that cross an odd number of times in D . Lemma 2.2.
For every semisimple drawing D of K n we have ocr( D ) = 2 ⌊ n/ ⌋− X k =0 E ≤≤ k ( D ) − (cid:18) n (cid:19) (cid:22) n − (cid:23) −
12 (1 + ( − n ) E ≤≤⌊ n/ ⌋− ( D ) . We recall that a face of a drawing D in the plane is a connected component of thecomplement of all the edges and vertices of D in R . The outer face of D is the unboundedface of D . Proof (sketch).
We just sketch the main idea, which is common with the proof of Lemma 2.1,and then explain the generalization to semisimple drawings. For the details, we refer thereader to [1, Theorem 1 and Proposition 1].Let D be a semisimple drawing of K n . A separation in D is an unordered triple { ab, c, d } ,where ab is an edge of D , c, d are vertices of D distinct from a, b , and the orientations of thetwo triangles abc and abd are opposite. Observe that { ab, c, d } is a separation in D if andonly if ab is a 1-edge (and also a halving edge ) in the complete subgraph of D induced by thevertices a, b, c, d . The total number of separations in D relates to both the crossing numberand the numbers of k -edges in the following way.(i) Every k -edge belongs to exactly k ( n − k −
2) separations.(ii) Every 4-tuple of vertices inducing a crossing contributes two separations, and every4-tuple of vertices inducing a planar drawing of K contributes three separations. Inparticular, for every complete subgraph D with 4 vertices we have the equality cr( D ) + E ( D ) = 3.Fact (i) is a direct consequence of the definitions. Fact (ii) is easily seen by inspecting allthree homeomorphism classes of simple drawings of K in the plane: there is one class withno crossing, and two classes with one crossing, which would form just one class on the sphere;see Figure 2. Lemma 2.1 follows from the facts (i) and (ii) by elementary computations.To generalize Lemma 2.1 to semisimple drawings, we observe that semisimple drawingsof K can be classified analogously as the simple drawings of K . In particular, the followingclaim implies that the equality ocr( D ) + E ( D ) = 3 is still satisfied for every semisimpledrawing D of K . 5igure 2: The three homeomorphism classes of simple drawings of K . The fat edges are1-edges. b cad a bcdF a,d F b,c Figure 3: Illustration to the proof of Lemma 2.2.
Claim.
A semisimple drawing D of K has at most one pair of edges crossing an odd numberof times. Moreover, D has three separations if ocr( D ) = 0 and two separations if ocr( D ) = 1 . In the rest of the proof we prove the claim. Let D be a semisimple drawing of K .Suppose that ocr( D ) = 0. Let abc be a triangle in D and let d be the fourth vertex of D . SeeFigure 3, left. If the edge da crosses bc , then either d and b share no face in the drawing of thesubgraph with edges ab, bc, ad , or d and c share no face in the drawing of the subgraph withedges ac, bc, ad . This means that one of the edges bd or cd either crosses an adjacent edge orcrosses another edge an odd number of times. Therefore, the edge da has no crossing withthe triangle abc . Analogous argument for the edges db and dc shows that D has no crossingsat all. In particular, D has three separations; see Figure 2, left.Now suppose that ocr( D ) ≥ ac and bd be two edges that cross an odd number oftimes. Since all the other edges are adjacent to both ac and bd , the vertices a, b, c, d share acommon face F in the drawing of the subgraph with edges ac, bd . Moreover, the cyclic orderof the vertices along the boundary of F is a, b, c, d , either clockwise or counter-clockwise. SeeFigure 3, right.We show that at most one more pair of edges can cross, either ab and cd , or ad and bc ,but only an even number of times. For example, in the drawing of the subgraph with edges ac , bd , ab , the vertices c and d belong to the same face, and the edge cd is allowed to crossonly the edge ab , each time switching faces. If ab and cd cross, then a and d share a uniqueface F a,d in the drawing of the graph K with edges ac, bd, ab, cd , and c and b share a uniqueface F b,c different from F a,d . Since the edges ad and bc are adjacent to all edges of K , theedge ad lies completely in F a,d , the edge bc lies completely in F b,c and thus ad and bc cannotcross. A symmetric argument shows that if ab and cd are disjoint, then ad and bc are eitherdisjoint or cross an even number of times. In any case, we have ocr( D ) ≤ D is at most 2).It remains to show that every semisimple drawing D of K with ocr( D ) = 1 has exactlytwo 1-edges. More precisely, we show that the two 1-edges always form a perfect matching.6 bcd a bcd Figure 4: An edge flip of ab .Let e be an edge in D incident with the outer face. An edge flip is an operation where theportion of e incident with the outer face is redrawn along the other side of the drawing; seeFigure 4. For drawings on the sphere, the edge flip is just a homeomorphism of the sphere.For every bounded face F of D , there is a sequence of edge flips that makes F the outer face.If D is a semisimple drawing of K , then every edge flip of an edge e changes the orientationof the two triangles adjacent to e . Consequently, exactly the four edges adjacent to e , forminga 4-cycle, change from 1-edges to 0-edges or vice versa. Also observe that the edge flip of e can be performed only if e is a 0-edge. It follows that 1-edges form a perfect matching in D if and only if they form a perfect matching in the drawing obtained by the edge flip.Let D be a semisimple drawing of K with ocr( D ) = 1. Let ac and bd be the two edgesthat cross an odd number of times. By performing edge flips, we may assume that all thevertices are adjacent to the outer face of the drawing of the subgraph H with edges ac and bd . Each edge e of the remaining four edges can be drawn in two essentially different wayswith respect to H , which differ just by an edge flip of e in H + e ; see Figure 4. In total,there are 16 possible combinations. We cannot, however, assume any particular combination,since not all edge flips are always available. Observe that the orientations of all triangles aredetermined by the four binary choices for the edges ab, bc, cd, ad . Also, changing the choicefor one edge e has the same effect on the orientations of the triangles as the edge flip of e .For one particular choice, for example the one yielding the middle drawing in Figure 2, the1-edges form a perfect matching. Changing the choice for a subset of edges yields either aperfect matching of 1-edges or a complete graph of 1-edges. However, the latter option isexcluded by the fact that in every semisimple drawing the edges incident with the outer faceare 0-edges. This finishes the proof of the claim and the lemma.Considering ≤ k -edges, ´Abrego and Fern´andez-Merchant [6] and Lov´asz et al. [28] provedthat for rectilinear drawings of K n , the inequality E ≤ k ≥ (cid:0) k +22 (cid:1) together with (2) givescr( G ) ≥ Z ( n ). However, there are simple x -monotone (even 2-page) drawings of K n where E ≤ k < (cid:0) k +22 (cid:1) for k = 1 [1]. ´Abrego et al. [1] showed that the inequality E ≤≤ k ≥ (cid:0) k +33 (cid:1) , whichis implied by inequalities E ≤ j ≥ (cid:0) j +22 (cid:1) for j ≤ k , is satisfied by all 2-page book drawings.We show that the same inequality is satisfied by all x -monotone semisimple drawings of K n .Let { v , v , . . . , v n } be the vertex set of K n . Note that we can assume that all verticesin an x -monotone drawing lie on the x -axis. We also assume that the x -coordinates of thevertices satisfy x ( v ) < x ( v ) < · · · < x ( v n ).The following observation describes the structure of k -edges incident to vertices on theouter face in semisimple drawings of complete graphs. See Figure 5, left.7 v v i k + 2 − ik + 2 − i v n Figure 5: Left: k -edges incident with a vertex on the outer face. Right: After removing v n ,at least k + 2 − i right edges at v i are invariant ≤ k -edges. Observation 2.3.
Let D be a semisimple drawing of K n , not necessarily x -monotone. Let v be a vertex incident to the outer face of D and let γ i be the i th edge incident to v inthe counter-clockwise order so that γ and γ n − are incident to the outer face in a smallneighborhood of v . Let v k i be the other endpoint of γ i . Then for every i, j , ≤ i < j ≤ n − ,the triangle v k i vv k j is oriented clockwise. Consequently, for every k with ≤ k ≤ ( n − / ,the edges γ k and γ n − k are ( k − -edges. For an x -monotone drawing D of K n , we use Observation 2.3 for the vertex v n and thedrawing D and then for each i , for the vertex v i and the drawing of the subgraph induced by v i , v i +1 , . . . , v n .The following definitions were introduced by ´Abrego et al. [1] for 2-page book drawings.Let D be a semisimple x -monotone drawing of K n and let D ′ be the drawing obtained from D by deleting the vertex v n together with its adjacent edges. A k -edge in D is a ( D, D ′ ) -invariant k -edge if it is also a k -edge in D ′ . It is easy to see that every ≤ k -edge in D ′ is alsoa ≤ ( k + 1)-edge in D . If 0 ≤ j ≤ k ≤ ⌊ n/ ⌋ −
1, then a (
D, D ′ )-invariant j -edge is called a( D, D ′ ) -invariant ≤ k -edge . Let E ≤ k ( D, D ′ ) denote the number of ( D, D ′ )-invariant ≤ k -edges.For i < j , the edge v i v j is called a right edge at v i . The right edges at v i have a naturalvertical order, which coincides with the order of their crossings with an arbitrary verticalline separating v i and v i +1 . The set of j topmost ( bottommost ) right edges at v i is the setof j right edges at v i that are above (below, respectively) all other right edges at v i in theirvertical order. Lemma 2.4.
Let D be a semisimple x -monotone drawing of K n and let k be a fixed integersuch that ≤ k ≤ ( n − / . For every i ∈ { , , . . . , k + 1 } , the k + 2 − i bottommost and the k + 2 − i topmost right edges at v i are ≤ k -edges in D . Moreover, at least k + 2 − i of these ≤ k -edges are ( D, D ′ ) -invariant ≤ k -edges.Proof. See Figure 5, right. The first part of the lemma follows directly from Observation 2.3.If the edge v i v n is one of the k + 2 − i topmost right edges at v i , then the k + 2 − i bottommostright edges at v i are ( D, D ′ )-invariant ≤ k -edges. Otherwise the k + 2 − i topmost right edgesat v i are ( D, D ′ )-invariant ≤ k -edges. Corollary 2.5.
We have E ≤ k ( D, D ′ ) ≥ k +1 X i =1 ( k + 2 − i ) = (cid:18) k + 22 (cid:19) . ≤≤ k -edges. The proof isessentially the same as in [1], we only extracted Lemma 2.4, which needed to be general-ized. Together with Lemma 2.2, Theorem 2.6 yields the second and the third equality inTheorem 1.1, by the same computation as in [1]. Theorem 2.6.
Let n ≥ and let D be a semisimple x -monotone drawing of K n . Then forevery k satisfying ≤ k < n/ − , we have E ≤≤ k ( D ) ≥ (cid:0) k +33 (cid:1) . Proof.
The proof proceeds by induction on n and k starting at n = 3 and k = −
1. The case n = 3 is trivially true, and the case k = − E ≤≤− ( D ) = 0 forevery drawing D . Let n ≥ D be a semisimple x -monotone drawing of K n . For theinduction step we remove the point v n together with its adjacent edges to obtain a drawing D ′ of K n − , which is also semisimple and x -monotone.Using Observation 2.3 we see that, for 0 ≤ i ≤ k < n/ −
1, there are two i -edges adjacentto v n in D and together they contribute with 2 P ki =0 ( k + 1 − i ) = 2 (cid:0) k +22 (cid:1) to E ≤≤ k ( D ) by (1).Let γ be an i -edge in D ′ . If i ≤ k , then γ contributes with ( k − i ) to the sum E ≤≤ k − ( D ′ ) = k − X i =0 ( k − i ) E i ( D ′ ) . We already observed that γ is either an i -edge or an ( i + 1)-edge in D . If γ is also an i -edge in D (that is, γ is a ( D, D ′ )-invariant i -edge), then it contributes with ( k + 1 − i ) to E ≤≤ k ( D ).This is a gain of +1 towards E ≤≤ k − ( D ′ ). If γ is an ( i + 1)-edge in D , then it contributesonly with ( k − i ) to E ≤≤ k ( D ). Therefore we have E ≤≤ k ( D ) = 2 (cid:18) k + 22 (cid:19) + E ≤≤ k − ( D ′ ) + E ≤ k ( D, D ′ ) . By the induction hypothesis we know that E ≤≤ k − ( D ′ ) ≥ (cid:0) k +23 (cid:1) and thus we obtain E ≤≤ k ( D ) ≥ (cid:18) k + 33 (cid:19) − (cid:18) k + 22 (cid:19) + E ≤ k ( D, D ′ ) . The theorem follows by plugging the lower bound from Corollary 2.5.
Here we finish the proof of Theorem 1.1 by showing that allowing adjacent edges to crossevenly yields no substantially new monotone drawings of K n .The rotation at a vertex v in a drawing is the clockwise cyclic order of the neighbors of v in which the corresponding edges appear around v . The rotation system of a drawing is theset of rotations of all its vertices. Proposition 2.7.
Let D be a weakly semisimple monotone drawing of K n . Then there is asemisimple monotone drawing D ′ of K n such that for every two edges e, f of K n , the parityof the number of crossings between e and f in D ′ is the same as in D . Moreover, D ′ and D have the same rotation system and the same above/below relations of vertices and edges. f efv wx x Figure 6: Left: the edge vw is forced to cross e or f an odd number of times. Right: decreasingthe total number of crossings. Proof.
Let O ( D ) be the set of pairs of edges of K n that cross an odd number of times in D . Let D ′ be a weakly semisimple monotone drawing of K n with minimum total number ofcrossings such that D ′ is strongly equivalent to D ′ , that is, D ′ and D have the same rotationsystem, the same above/below relations of vertices and edges and O ( D ′ ) = O ( D ). We showthat D ′ is semisimple.Suppose for contrary that D ′ has two adjacent edges e, f that cross. Since D ′ is weaklysemisimple, e and f cross at least twice. Let v be the common vertex of e and f and supposethat e is above f in the neighborhood of v . Let x and x be the two crossings of e and f closest to v . See Figure 6, left. Let B be the closed topological disc bounded by the twoportions of e and f between x and x . Clearly, B has no vertex on its boundary. Moreover,we claim that B has no vertex in its interior. For if B contains a vertex w in its interior,then w is below f and above e . This implies that the edge vw is below f and above e in theneighborhood of v , which is absurd.Since B contains no vertices, every edge other than e and f crosses the boundary of B an even number of times. Therefore, by redrawing an open segment of e or f containing x and x along the other side of B , we obtain a drawing strongly equivalent to D ′ with at mostcr( D ′ ) − D ′′ stronglyequivalent to D such that for every two edges, the number of their common crossings in D ′′ is not larger than in D .By Proposition 2.7, the odd crossing number of a weakly semisimple monotone drawingof K n is equal to the odd crossing number of some semisimple monotone drawing of K n . Thisproves the first equality in Theorem 1.1. In this section we develop a combinatorial characterization of x -monotone drawings basedon the signature functions introduced by Peters and Szekeres [40] as generalizations of ordertypes of planar point sets. Let T n be the set of ordered triples ( i, j, k ) with i < j < k , of theset [ n ] = { , , . . . , n } and let Σ n be the set of signature functions σ : T n → {− , + } . The set T n may be also regarded as the set (cid:0) [ n ]3 (cid:1) of all unordered triples, since we write all the triplesin the increasing order of their elements.Let D be an x -monotone drawing of the complete graph K n = ( V, E ) with vertices v , v , . . . , v n such that their x -coordinates satisfy x ( v ) < x ( v ) < · · · < x ( v n ). We as-sign a signature function σ ∈ Σ n to the drawing D according to the following rule. For everyedge e = v i v k ∈ E and every integer j ∈ ( i, k ), let σ ( i, j, k ) = − if the point v j lies above10 i v j v k v i v j v k σ ( i, j, k ) = + σ ( i, j, k ) = − Figure 7: The negative and the positive signature σ ( i, j, k ).the arc representing the edge e and σ ( i, j, k ) = + otherwise. See Figure 7. Note that if thedrawing D is also semisimple, then a triangle v i v j v k , with j ∈ ( i, k ), is oriented clockwise(counter-clockwise) if and only if σ ( i, j, k ) = − ( σ ( i, j, k ) = +, respectively).It is easy to see that, for every signature function σ ∈ Σ n , there exists an x -monotonedrawing D which induces σ . However, some signature functions are induced only by drawingsthat are not semisimple. We show a characterization of simple and semisimple x -monotonedrawings by small forbidden configurations in the signature functions.For integers a, b, c, d ∈ [ n ] with a < b < c < d , signs ξ , ξ , ξ , ξ ∈ {− , + } and a signaturefunction σ ∈ Σ n , we say that the 4-tuple ( a, b, c, d ) is of the form ξ ξ ξ ξ in σ if σ ( a, b, c ) = ξ , σ ( a, b, d ) = ξ , σ ( a, c, d ) = ξ , and σ ( b, c, d ) = ξ . Alternatively, we write σ ( { π ( a ) , π ( b ) , π ( c ) , π ( d ) } ) = ξ ξ ξ ξ for any permutation π of the set { a, b, c, d } .For a sign ξ ∈ {− , + } we use ξ to denote the opposite sign, that is, if ξ = + then ξ = − and conversely, if ξ = − then ξ = +. x -monotone drawings Theorem 3.1.
A signature function σ ∈ Σ n can be realized by a semisimple x -monotonedrawing if and only if every -tuple of indices from [ n ] is of one of the forms ++++ , −−−− , ++ −− , −− ++ , − ++ − , + −− + , −−− + , +++ − , + −−− , − +++ in σ . The signature function σ can be realized by a simple x -monotone drawing if, in addition,there is no -tuple ( a, b, c, d, e ) with a < b < c < d < e such that σ ( a, b, e ) = σ ( a, d, e ) = σ ( b, c, d ) = σ ( a, c, e ) . See Figure 13 and Figure 10 for an illustration of the first and the second part of thetheorem.
Proof.
Let σ be a signature function with a forbidden -tuple , that is, an ordered 4-tuple( a, b, c, d ) whose form is not listed in the statement of the theorem. Such a 4-tuple ( a, b, c, d )is one of the forms ξ ξ ξ ξ or ξ ξ ξ ξ where ξ , ξ ∈ {− , + } . If ( a, b, c, d ) is of the form+ − + ξ where ξ ∈ {− , + } is an arbitrary sign, then the edges v a v c and v a v d are forced to crossbetween the vertical lines going through v b and v c ; see Figure 8. But this is not allowed in asemisimple drawing and we have a contradiction. The other cases are symmetric.11 b c d + + − Figure 8: A 4-tuple ( a, b, c, d ) of the form + − + ξ forces two adjacent edges to cross.On the other hand, let σ be a signature function such that every 4-tuple is of one of theten allowed forms in σ . We will construct a semisimple x -monotone drawing D of K n whichinduces σ . We use the points v i = ( i, i ∈ [ n ], as vertices and connect consecutive pairs ofvertices by straight-line segments.For m ∈ [ n ], let L m be the vertical line containing v m . In every x -monotone drawing,the line L m intersects every edge { v i , v j } with 1 ≤ i < m ≤ j ≤ n exactly once. To drawthe edges of K n , it suffices to specify the positions of their intersections with the lines L m and to draw the edges as polygonal lines with bends at these intersections. Instead of theabsolute position of these intersections on L m , we only need to determine their vertical totalordering, which we represent by a total ordering ≺ m of the corresponding edges. The edgeswhose right endpoint is v m will be ordered by ≺ m according to their vertical order in the leftneighborhood of v m . The edges with left endpoint v m are not considered in ≺ m .The idea of the construction is to interpret the signature function as the set of above/belowrelations for vertices and edges and take a set of orderings ≺ m that obey these relations andminimize the total number of crossings. In the rest of the proof we show a detailed, explicitconstruction of the orderings ≺ m which induce an x -monotone semisimple drawing.For i ∈ [ n ], we define an ordering ⋖ i of the edges with a common left endpoint v i (that is,the right edges at v i ) in the following way. If e = { v i , v j } and f = { v i , v k } , i < j, k , are twosuch edges, then we set e ⋖ i f if either j < k and σ ( i, j, k ) = +, or k < j and σ ( i, k, j ) = − .Clearly, the relation ⋖ i is irreflexive, antisymmetric and for every two right edges e, f at v i either e ⋖ i f or f ⋖ i e . To show that ⋖ i is a total ordering, it remains to prove that it istransitive. Suppose for contrary that there are three edges e = { v i , v j } , f = { v i , v k } and g = { v i , v l } with i < j < k < l such that e ⋖ i f , f ⋖ i g and g ⋖ i e . Then σ ( i, j, k ) = +, σ ( i, k, l ) = + and σ ( i, j, l ) = − , so the 4-tuple i, j, k, l is of the form + − + ξ , which is forbidden.Similarly, if f ⋖ i e , e ⋖ i g and g ⋖ i f , then the 4-tuple i, j, k, l is of the form − + − ξ , which isforbidden as well.We proceed by induction on m . In the case m = 1 the ordering ≺ is empty. For m = 2the ordering ≺ compares only edges with the common endpoint v , so we can set ≺ = ⋖ .Since all the edges are drawn by line segments starting in a common endpoint, no crossingsappear between L and L .Let m >
2. For the inductive step we consider the following sets S , . . . , S of edges whichintersect L m − and L m (see Figure 9): 12 m − v m S S S S S S L m L m − Figure 9: Placing edges and minimizing the number of crossings. S = {{ v i , v j } | σ ( i, m − , j ) = − , σ ( i, m, j ) = −} ,S = {{ v m − , v j } | σ ( m − , m, j ) = −} ,S = {{ v i , v j } | σ ( i, m − , j ) = + , σ ( i, m, j ) = − or j = m } ,S = {{ v i , v j } | σ ( i, m − , j ) = − , σ ( i, m, j ) = + or j = m } ,S = {{ v m − , v j } | σ ( m − , m, j ) = + } ,S = {{ v i , v j } | σ ( i, m − , j ) = + , σ ( i, m, j ) = + } . The edges within sets S and S are ordered according to ⋖ m − and the edges in each of theremaining sets S k according to ≺ m − . For e ∈ S k and f ∈ S l where k < l , we set e ≺ m f .Observe that ≺ m is a total ordering.We show that the drawing D determined by the orders ≺ m is semisimple. Suppose forcontradiction that two adjacent edges e = { v i , v j } and f = { v i , v k } , with i < j, k and e ⋖ i f ,cross. Their leftmost crossing occurs between lines L m − and L m , where i < m − m ≤ j, k . There are three cases:(i) e ∈ S and f ∈ S ,(ii) e ∈ S and f ∈ S , or(iii) e ∈ S and f ∈ S .We analyze the cases (i) and (iii) together, case (i) and case (ii) are symmetric. If j < k then σ ( i, m, k ) = − and by the definition of the relation ⋖ i , we have σ ( i, j, k ) = +. Thisfurther implies that m < j and σ ( i, m, j ) = +. Thus ( i, m, j, k ) forms a forbidden 4-tuple. If k < j , then σ ( i, m, j ) = +, σ ( i, k, j ) = − , which implies that m < k and σ ( i, m, k ) = − , andso we obtain a forbidden 4-tuple ( i, m, k, j ).Now suppose that two adjacent edges e = { v i , v k } and f = { v j , v k } , with i, j < k , cross.Their leftmost crossing occurs between lines L m − and L m , where i, j ≤ m − m < k .We may assume that f ≺ m e and e ≺ m − f . There are five cases:13 b c d + + − e + Figure 10: A forbidden 5-tuple ( a, b, c, d, e ) forces at least two crossings between v a v e and v b v d .(i) e ∈ S and f ∈ S ,(ii) e ∈ S and f ∈ S ,(iii) e ∈ S and f ∈ S ,(iv) e ∈ S and f ∈ S , or(v) e ∈ S and f ∈ S .Case (i) and case (ii) are symmetric, as well as case (iv) and case (v). Therefore it issufficient to consider cases (i), (iii) and (v). In all these three cases σ ( j, m, k ) = − and σ ( i, m, k ) = +. If j < i , then σ ( j, i, k ) = + since e ≺ m − f and the edges e and f do not crossto the left of L m − . Hence ( j, i, m, k ) forms a forbidden 4-tuple. If i < j , then analogously σ ( i, j, k ) = − and ( i, j, m, k ) forms a forbidden 4-tuple. This finishes the proof that D issemisimple.It remains to show the second part of the theorem. If D is a drawing with a signaturefunction σ with a forbidden -tuple ( a, b, c, d, e ), then D is not simple as the edges v a v e and v b v d are forced to cross at least twice; see Figure 10.In the rest of the proof we show the second part of the theorem.Given a signature function σ with no forbidden 4-tuples and 5-tuples we apply the sameconstruction as before to obtain a semisimple x -monotone drawing D . We show that D is, inaddition, simple. Since D is semisimple, no two crossing edges have an endpoint in common.By the construction of D , every crossing c of two edges e and f occurs between lines L m and L m +1 for some m ∈ [ n −
1] and we say that v m +1 is the right neighbor of c . The rightneighbor is either an endpoint of e or f or it separates the crossings of L m +1 with e and f .Suppose that there are edges e = v i v j and f = v k v l with i < k < j, l that cross at least twice.We show that then there is always a forbidden 4-tuple or a forbidden 5-tuple in σ .Let v m be the right neighbor of the leftmost crossing and v m ′ the right neighbor of thesecond leftmost crossing of e and f . Observe that i, k < m < m ′ ≤ j, l .First assume that l < j . Refer to Figure 11. If σ ( i, k, j ) = σ ( i, l, j ) = ξ for some ξ ∈ {− , + } , then ξ = σ ( k, m, l ) = σ ( i, m, j ) and so ( i, k, m, l, j ) forms a forbidden 5-tuple. If σ ( i, k, j ) = σ ( i, l, j ) = ξ for some ξ ∈ {− , + } , then e and f cross at least three times and so m ′ < l, j . We have ξ = σ ( k, m, l ) = σ ( i, m, j ) = σ ( k, m ′ , l ) = σ ( i, m ′ , j ). If σ ( k, m, m ′ ) = ξ ,then ( k, m, m ′ , l ) forms a forbidden 4-tuple. If σ ( k, m, m ′ ) = ξ , then ( i, k, m, m ′ , j ) forms aforbidden 5-tuple.Conversely let j < l . Refer to Figure 12. Assume that σ ( i, k, j ) = σ ( k, j, l ) = ξ for some ξ ∈ {− , + } . Then ξ = σ ( k, m, l ) = σ ( i, m, j ). If σ ( k, m, j ) = ξ , we get a forbidden 4-tuple( i, k, m, j ), otherwise σ ( k, m, j ) = ξ and we get a forbidden 4-tuple ( k, m, j, l ). Finally, assume14 k m l + j + i k m l + j − m ′ Figure 11: Edges v i v j and v k v l crossing twice imply a forbidden 5-tuple or 4-tuple; case l < j . i k m l + j + i k m l + j − m ′ Figure 12: Edges v i v j and v k v l crossing twice imply a forbidden 5-tuple or 4-tuple; case j < l .15 +++ −−−− ++ −−−− ++ −−− ++ −−−− ++++++ − pseudolinear + −− + − ++ − semisimple Figure 13: The 4-tuples in pseudolinear and semisimple drawings.that σ ( i, k, j ) = σ ( k, j, l ) = ξ for some ξ ∈ {− , + } . The proof in this case is identical to theproof of the case l < j and σ ( i, k, j ) = σ ( k, j, l ) = ξ in the previous paragraph. x -monotone drawings A drawing D of a complete graph K n is pseudolinear (also pseudogeometric or extendable )if the edges of D can be extended to unbounded simple curves that cross each other exactlyonce, thus forming an arrangement of pseudolines . The vertices of D together with the (cid:0) n (cid:1) pseudolines extending the edges are said to form a pseudoarrangement of points (also generalized configuration of points ). Note that the pseudoarrangement of points extending D is usually not unique as there is a certain freedom in choosing where the pseudolines extendingdisjoint noncrossing edges of D cross.It is well known that every arrangement of pseudolines can be made x -monotone by asuitable isotopy of the plane (this follows, for example, by the duality transform established byGoodman [18, 20]). Therefore, every pseudolinear drawing of K n is isotopic to an x -monotonepseudolinear drawing. Every rectilinear drawing of K n is x -monotone and pseudolinear, butthere are pseudolinear drawings of K n that cannot be “stretched” to rectilinear drawings.We show that x -monotone pseudolinear drawings of K n can be characterized in a combina-torial way by forbidden 4-tuples in the corresponding signature function, by further restrictingthe conditions on the signatures in Theorem 3.1. In fact, the conditions in Theorem 3.2 areprecisely the geometric constraints that Peters and Szekeres [40] used to restrict the set ofsignature functions in their investigation of the Erd˝os–Szekeres problem. Figure 13 illustratesthe classification of 4-tuples from Theorem 3.1 and Theorem 3.2. Theorem 3.2.
A signature function σ ∈ Σ n can be realized by a pseudolinear x -monotonedrawing if and only if every ordered -tuple of indices from [ n ] is of one of the forms ++++ , +++ − , ++ −− , + −−− , −−−− , −−− + , −− ++ , − +++16 n σ . Pseudolinear drawings of complete graphs are equivalent to
CC systems introduced byKnuth [26], although this equivalence is not easily seen. The CC systems are ternary counter-clockwise relations of finite sets satisfying a certain set of five axioms involving triples, 4-tuples or 5-tuples of elements. CC systems generalize the order types of planar point setsin general position: an ordered triple in the counter-clockwise relation is interpreted as atriple of points in the plane placed in the counter-clockwise order, like a triple with signature+ in the signature function. Unlike the signature functions, the CC systems have no fixedordering of the elements. Therefore, some of the axioms for CC systems involve 5-tuples ofelements, whereas 4-tuples are sufficient in the case of signature functions. In fact, the axiomsof CC systems specify exactly that every 5-tuple of elements can be realized as a point setin the plane. Knuth [26] established a correspondence between CC systems and reflectionnetworks (also called wiring diagrams ), which are simple arrangements of pseudolines dual tothe pseudoarrangements of points extending the pseudolinear drawings of complete graphs.Knuth [26] also showed a two-to-one correspondence between CC systems and uniform acyclicoriented matroids of rank 3 on the same underlying set. Here the CC system is, in fact, the chirotope of the corresponding oriented matroid.Streinu [39] characterized sets of signed circular permutations ( directed clusters of stars )that arise from generalized configurations of n points as circular sequences of pseudolines ateach of the n points, and provided an O ( n ) drawing algorithm, partially similar to ours. Itis easy to show that the set of signed circular permutations determines the orientation of alltriangles (and thus the corresponding CC system) and vice versa. However, many details areomitted in the extended abstract [39].Felsner and Weil [14, 15] proved that triangle-sign functions of simple arrangements of n pseudolines are precisely those functions f : (cid:0) [ n ]3 (cid:1) → { + , −} that are monotone on all 4-tuples.This is the same condition as the condition on signature functions in Theorem 3.2. That is,Theorem 3.2 is a dual analogue of Felsner’s and Weil’s result. Felsner and Weil [14, 15] alsointroduced r -signotopes , a notion unifying permutations, allowable sequences and monotonetriangle-sign functions of simple arrangements. In this notation, the signature functionssatisfying the conditions of Theorem 3.2 are 3-signotopes.Although Theorem 3.2 can be deduced from any of these previous results, we still believethat providing a direct, self-contained proof has its merit. Clearly, every pseudolinear x -monotone drawing of K is isotopic to one of the eight drawingsof K in the first two columns in Figure 13, and thus its signature function has one of thecorresponding eight forms.Let σ be a signature function such that every 4-tuple is of one of the eight allowed formsin σ . We show that there is a pseudolinear x -monotone drawing D of K n which induces σ . Unlike in the proof of Theorem 3.1, we do not provide an explicit construction of thedrawing. However, our proof can be easily transformed into a polynomial algorithm findingsuch a drawing.Again, we use the points v i = ( i, i ∈ [ n ], as vertices. For m ∈ [ n ], let L m be the verticalline containing v m . Let L be the vertical line containing the point (0 , K n and the pseudolines extending the edges, up to a combi-natorial equivalence, it suffices to specify the left and right vertical orders of the pseudolines17 i v i L i +1 v i +1 ≺ Ri ≺ Li +1 L n v n ≺ Rn L v ≺ L L ( ≺ Rn ) − Figure 14: Drawing the curves p i,j . ≺ Lm ≺ Rm p m,j p m,k L m v m ≺ Lm ≺ Rm p m,j p i,j L m v m ≺ Lm ≺ Rm p k,l p i,j L m v m Figure 15: Illustration of the binding conditions for the orders ≺ Lm and ≺ Rm .crossing at each of the points v i , and the relative positions of the intersections of the pseudo-lines with the lines L , L , . . . , L n .For i, j ∈ [ n ] , i = j, let p i,j be the pseudoline extending the edge v i v j . We emphasizethat we use both p i,j and p j,i to denote the same pseudoline. We draw the pseudolines inthe following way; see Figure 14. For every i ∈ [ n ], we draw a portion of each pseudoline p i,j containing v i as two short segments joining points v i − ( ε, δ j ), v i and v i + ( ε, δ ′ j ), where ε, δ j and δ ′ j are sufficiently small and the relative order of the y -coordinates δ j ( δ ′ j ) is consistentwith the left (right, respectively) vertical order of the pseudolines at v i . It will follow fromthe construction that we can take δ ′ j = − δ j , so the two segments actually form one segmentwith midpoint v i . We also choose the intersection points of the pseudolines p i,j with the lines L m ( i, j = m ) sufficiently far from the points v m and consistently with the relative positionsspecified. Then for each pseudoline p i,j , we connect consecutive intersections with lines L m , m ∈ { , , . . . , n } \ { i, j } , and points v i ± ( ε, δ j ) and v j ± ( ε, δ i ) by straight-line segments.Finally, we attach horizontal rays starting at intersections of p i,j with L directed to the left,and similarly, horizontal rays starting at intersections of p i,j with L n (if i, j = n ) and at points v n + ( ε, δ ′ j ), directed to the right.We represent the order of intersections of the pseudolines with a vertical line by theorder of the corresponding pseudolines. For m ∈ [ n ], we define two total orders ≺ Lm and ≺ Rm on the set P = { p i,j ; 1 ≤ i < j ≤ n } of all pseudolines. The order ≺ Lm ( ≺ Rm ) representsthe vertical order of the pseudolines in the left (right, respectively) neighborhood of L m .See Figure 15. We require the two orders ≺ Lm and ≺ Rm to be mutually inverse on the set P m = { p i,j ; 1 ≤ i < j ≤ n, m ∈ { i, j }} of pseudolines containing v m and identical otherwise,that is, • p m,k ≺ Lm p m,j if and only if p m,j ≺ Rm p m,k , for all j, k ∈ [ n ] such that j, k, m are distinct,and 18 i v j v k p i,j p i,k p j,k Figure 16: Three pseudolines determined by v i , v j and v k . • p i,j ≺ Lm p k,l if and only if p i,j ≺ Rm p k,l , for all i, j, k, l ∈ [ n ] such that i < j , k < l , { i, j } 6 = { k, l } , and { i, j } ∩ { k, l } ∩ { m } = ∅ .We also define a total order ≺ on P as follows: • ≺ ≡ ( ≺ Rn ) − .That is, ≺ is the inverse of ≺ Rn . The objective here is to make every two pseudolinescross an odd number of times (in particular, at least once).Further conditions on the orders ≺ Lm and ≺ Rm are determined by the signature function σ ;see Figure 16. For i ∈ [ n ], we fix the orders ≺ Li and ≺ Ri on P i as total orders in the followingway. For all j, k ∈ [ n ] such that i = j < k = i , • if i < j < k , then p i,j ≺ Ri p i,k if σ ( i, j, k ) = + and p i,k ≺ Ri p i,j if σ ( i, j, k ) = − , • if j < k < i , then p i,j ≺ Ri p i,k if σ ( j, k, i ) = + and p i,k ≺ Ri p i,j if σ ( j, k, i ) = − , • if j < i < k , then p i,j ≺ Ri p i,k if σ ( j, i, k ) = + and p i,k ≺ Ri p i,j if σ ( j, i, k ) = − .To show that ≺ Ri and ≺ Li are total orders on P i , we need to verify the transitivity of ≺ Ri .Suppose for contrary that p i,j ≺ Ri p i,k ≺ Ri p i,l ≺ Ri p i,j for some j, k, l ∈ [ n ] \ { i } . We mayassume that j < k, l . In the following table we list the eight cases of σ ( { i, j, k, l } ) accordingto the relative order of i, j, k, l . The symbol ξ stands for a sign that is not determined.order σ ( { i, j, k, l } ) order σ ( { i, j, k, l } ) i < j < k < l + − + ξ i < j < l < k − + − ξj < k < l < i ξ + − + j < l < k < i ξ − + − j < i < k < l + − ξ + j < i < l < k − + ξ − j < k < i < l + ξ − + j < l < i < k − ξ + − It follows that in every relative ordering, the indices i, j, k, l form a forbidden 4-tuple.Therefore, both ≺ Ri and ≺ Li are transitive on P i .For every i, j, k ∈ [ n ] such that i = j < k = i and for every p ∈ P i , we also fix the followingconditions: • if i < j < k , then p ≺ Ri p j,k if σ ( i, j, k ) = − and p j,k ≺ Ri p if σ ( i, j, k ) = +, • if j < k < i , then p ≺ Ri p j,k if σ ( j, k, i ) = − and p j,k ≺ Ri p if σ ( j, k, i ) = +,19 f ef efv i v j v i Figure 17: Bigons formed by curves e, f . Left: a minimal empty bigon. Middle: a smoothbigon. Right: a bigon that is neither smooth nor empty. • if j < i < k , then p ≺ Ri p j,k if σ ( j, i, k ) = + and p j,k ≺ Ri p if σ ( j, i, k ) = − .These conditions represent the above/below relations of the pseudolines p j,k and the points v i implied by σ (see Figure 7).It is easy to see that all the conditions required so far for the orders ≺ Lm , ≺ Rm and ≺ canbe simultaneously satisfied. For example, for crossings of L m , m ∈ [ n ], with the pseudolinesdisjoint with v m , the conditions only specify a partition of these pseudolines into two subsets:those crossing L m below v m and those crossing L m above v m .Finally, we choose total orders ≺ Lm , ≺ Rm and ≺ on P satisfying all the required conditionsand such that the total number of crossings of the pseudolines is minimized. Combinatorially,this last condition is equivalent to minimizing the total number of inversions between pairs ofpermutations corresponding to ≺ Ri and ≺ Li +1 , for all i ∈ [ n − ≺ Rn ) − and ≺ L .Let A be an arrangement of piecewise linear curves p i,j constructed from the total orders ≺ Lm , ≺ Rm and ≺ . Assume that no three curves from A cross at the same point, except for thepoints v , v , . . . , v n . We show that every two curves in A cross exactly once and thus deserveto be referred to as pseudolines.Let e, f be two x -monotone curves from the arrangement A . A bigon B formed by e and f is a closed topological disc bounded by two simple arcs e ′ , f ′ that have common endpointsand disjoint relative interiors, and such that e ′ is a portion of e and f ′ is a portion of f . Thecommon endpoints of e ′ and f ′ are the vertices of B .It will be convenient to consider A as an arrangement of curves on the M¨obius stripobtained from the infinite rectangle { ( x, y ) ∈ R ; 0 ≤ x ≤ n + ε, y ∈ R } by identifying eachpoint (0 , y ), y ∈ R , with the point ( n + ε, − y ). We extend the notion of a bigon to includealso special bigons that are bounded by portions of two curves from A in the M¨obius stripand intersect the line L . A bigon that does not intersect L is an ordinary bigon . Observethat if two curves e and f cross k times, then e and f form exactly k − B is empty if B ∩ { v , v , . . . , v n } = ∅ . A bigon B is smooth if the boundary of B does not intersect { v , v , . . . , v n } . See Figure 17. Claim 3.3.
No two curves from A form an empty bigon.Proof. Suppose that e and f are two curves from A that form an empty bigon B . Let e ′ ⊂ e and f ′ ⊂ f be the two arcs forming the boundary of B . Suppose further that B is inclusionminimal, among all pairs of pseudolines. Moreover, we may suppose that both e ′ and f ′ areinclusion minimal among all arcs forming the bottom or the top boundary of some bigon.Then every curve g from A distinct from e and f is either disjoint with B or crosses both20 ′ and f ′ exactly once. We can thus redraw e along f ′ outside B and decrease the totalnumber of crossings by two. After this operation, the resulting arrangement still satisfies allthe conditions specified by the orders ≺ Lm , ≺ Rm and ≺ , as the neighborhoods of the points v i and all the above/below relations of the pseudolines and points v i remain unaffected. Corollary 3.4.
Every smooth bigon formed by two curves from A contains at least one point v m in its interior. For i, j ∈ [ n ], i = j , let e i,j be the portion of the curve p i,j between the points v i and v j ,representing the edge v i v j of K n . Claim 3.5.
Every two curves from A sharing a point v i cross only at v i .Proof. Suppose that for some j, k ∈ [ n ] \ { i } , j < k , the curves p i,j and p i,k cross more thanonce. By symmetry, we may assume that i < k and p i,j ≺ Ri p i,k . If j < i , we may furtherassume that e i,j crosses p i,k at most as many times as e i,k crosses p i,j . We have five cases; seeFigure 18.(i) i < j and e i,j crosses e i,k . By the definition of ≺ Ri , we have σ ( i, j, k ) = +. Consequently, p i,k crosses L j above v j . This further implies that e i,j and e i,k cross at least twiceand thus they form a smooth ordinary bigon B . Let v m be a point in the interior of B guaranteed by Corollary 3.4. We have i < m < j < k , σ ( i, m, j ) = + and σ ( i, m, k ) = − ,which implies that σ ( i, m, j, k ) = + − + ξ for some ξ ∈ {− , + } .(ii) i < j and e i,k crosses p i,j but not e i,j . Again, we have σ ( i, j, k ) = +. Consequently, p i,k crosses L j above v j and p i,j crosses L k below v k . This further implies that p i,j and e i,k cross at least twice (not counting the point v i ) and thus they form a smoothordinary bigon with a point v m in its interior. Taking the leftmost such bigon, we have i < j < m < k and σ ( i, j, m, k ) = − + − ξ .(iii) i < j , e i,j does not cross p i,k and e i,k does not cross p i,j . In this case all three points v i , v j and v k are on the boundary of the same bigon B formed by p i,j and p i,k and only v i is a vertex of B . Since p i,j and p i,k cross at least three times, they form at least threebigons. At most two of the bigons contain v i , thus at least one of them, B ′ , is smoothand has a point v m in its interior. We may assume that B ′ shares a vertex with B . Notethat either of B and B ′ can be special, so we have two cases: m > k or m < i . In the firstcase we have σ ( i, j, k, m ) = + − + ξ , in the second case we have σ ( m, i, j, k ) = + − ξ +.(iv) j < i and e i,k crosses p i,j . Since v k lies above p i,j , the curves e i,k and p i,j cross at leasttwice and thus form a smooth bigon, containing a point v m in its interior. We have j < i < m < k and σ ( j, i, m, k ) = − + ξ − .(v) j < i , e i,j does not cross p i,k and e i,k does not cross p i,j . The curves p i,j and p i,k format least three bigons, but only two of them, B and B , contain v i . Let B be the bigonother than B sharing a vertex with B . By the assumptions, B ∩ { v i , v j , v k } = ∅ ,so B is smooth and contains some point v m . We have either m < j < i < k with σ ( m, j, i, k ) = + ξ − +, or j < i < k < m with σ ( j, i, k, m ) = + − ξ +.In every case there is a forbidden 4-tuple, which is a contradiction.21 m j ki mj ki mj k (i)(ii)(iii) im j k (iv)(v) i kki mj m Figure 18: Bigons with vertex at v i are forbidden.22 i k Figure 19: Smooth bigons are also forbidden. i ki k
Figure 20: The curve p ik has to cross p i,j or p k,l in some other point than v i or v k . Claim 3.6.
Let i, j, k, l ∈ [ n ] such that |{ i, j, k, l }| = 4 . Then p i,j and p k,l do not form asmooth bigon.Proof. Let B be a smooth bigon, containing a point v m in its interior. By Claim 3.5, wemay asume that p i,m and p i,j cross at most once. Therefore, p i,m enters and exits the bigon B through p k,l , perhaps more than once; see Figure 19. Similarly, p k,m enters and exits thebigon B through p i,j . Since p i,m and p k,m cross at v m and they enter and exit B throughopposite sides, they cross at least twice. This is in contradiction with Claim 3.5.If two curves from A cross more than once, then by Corollary 3.4, Claim 3.5 and Claim 3.6,they are of the form p i,j and p k,l with i, j, k, l distinct, and they form no smooth bigon.Therefore, every bigon formed by p i,j and p k,l contains at least one of the points v i , v j , v k , v l on its boundary, but not at the vertices. In particular, p i,j and p k,l form exactly three bigons.Suppose that v i and v k are on the boundary of two different bigons. Up to symmetry, we arein one of the cases depicted in Figure 20 (considering all symmetries of the annulus, thereis just one case). The curve p i,k has to cross p i,j or p k,l at least twice, which contradictsClaim 3.5. Similar argument for pairs v i , v l and v j , v k implies that all four points v i , v j , v k , v l are on the boundary of the same bigon. Therefore at least two bigons are smooth, whichcontradicts Claim 3.6. This finishes the proof of the theorem. A similar characterization of rectilinear drawings of K n (equivalently, order types of planarpoint sets in general position) in terms of signature functions or CC systems with a finitenumber of forbidden configurations is impossible: for example, Bokowski and Sturmfels [12]constructed infinitely many minimal CC systems (simplicial affine 3-chirotopes) that are notrealizable as sets of points in the plane. This and related results were also referred to by thephrase “missing axiom for chirotopes is lost forever”.23oreover, recognizing signature functions of rectilinear drawings of K n (or, order typesof planar point sets in general position), is polynomially equivalent to rectilinear realizabilityof complete abstract topological graphs and to stretchability of pseudoline arrangements [27],which is polynomially equivalent to the existential theory of the reals [29]. In the terminologyintroduced by Schaefer [36], these problems are ∃ R -complete. It is known that ∃ R -completeproblems are in PSPACE [13] and NP-hard, but they are not known to be in NP. x -monotone drawings Note that in a simple x -monotone drawing of K n , the crossings appear only between edgeswhose endpoints induce a 4-tuple of one of the forms ++++, −−−− , ++ −− , −− ++, − ++ − ,+ −− +. Analogously as for the rectilinear drawings of K n , we may call these 4-tuples convex .Then, for a simple x -monotone drawing D of K n the crossing number of D equals the numberof convex 4-tuples. A similar notion of convexity for general k -tuples was used by Peters andSzekeres [40].This description of crossings is convenient for computer calculations. Using it, we haveobtained a complete list of optimal x -monotone drawings of K n for n ≤
10. To enumerate“essentially different” drawings we used the following approach.Let D be an x -monotone drawing of K n which induces a signature function σ . We canassume that the vertices are points placed on the same horizontal line (the x -axis). Thefollowing operations on D and σ produce a signature function σ ′ of a simple monotone drawing D ′ that is homeomorphic to D on the sphere, by a homeomorphism that does not necessarilypreserve the labels of vertices. In some cases we just describe the transformation of thedrawing; the new signature function σ ′ can be then computed in a straightforward way.(a) Vertical reflection : setting σ ′ ( i, j, k ) = σ ( i, j, k ) for every ( i, j, k ) ∈ T n .(b) Horizontal reflection : setting σ ′ ( i, j, k ) = σ ( n + 1 − k, n + 1 − j, n + 1 − i ) for every( i, j, k ) ∈ T n .(c) Shifting v : if every edge incident to v lies completely above or completely below the x -axis, that is, σ (1 , i, k ) = σ (1 , j, k ) for every k ∈ { , . . . , n } and 1 < i, j < k , then wecan move v to the position of v n and move every v i +1 to the position of v i , for every1 ≤ i ≤ n − Switching consecutive points : let j ∈ [ n − ξ ∈ {− , + } such that σ ( j, j +1 , k ) = ξ for every j + 1 < k ≤ n and σ ( i, j, j + 1) = ξ for every 1 ≤ i < j , then we canswitch the positions of v j and v j +1 . After the switch, we have σ ′ ( j, j + 1 , k ) = ξ for every j + 1 < k ≤ n and σ ′ ( i, j, j + 1) = ξ for every 1 ≤ i < j .(e) Redrawing the edge v v n : in every crossing minimal x -monotone drawing, the edge v v n crosses no other edge, since we can always redraw this edge along the top or the bottompart of the boundary of the outer face. The signature function σ thus satisfies σ (1 , i, n ) = ξ for some ξ ∈ { + , −} and for every i , 1 < i < n . We may thus simultaneously change allthe signatures σ (1 , i, n ).We say that two x -monotone drawings D and D ′ are switching equivalent if there is asequence of operations (a)–(e) such that, when applied to D , we obtain a drawing which hasthe same signature function as D ′ . We have found representatives of all switching equivalence24igure 21: Left: a crossing minimal x -monotone drawing of K homeomorphic to the cylindri-cal drawing. Right: a crossing minimal x -monotone drawing of K that is not homeomorphicto a 2-page book drawing and neither to the cylindrical drawing.Number of vertices 5 6 7 8 9 10Number of drawings 1 1 5 3 510 38Table 1: Numbers of switching equivalence classes of crossing minimal x -monotone drawingsof K n for n ≤ x -monotone drawings of K n , for n ≤
10. Their numbers are givenin Table 1.´Abrego et al. [1] proved that for every even n , there is a unique crossing minimal 2-pagebook drawing of K n , up to a homeomorphism of the sphere. We have found crossing minimal x -monotone drawings of K and K that are not homeomorphic to 2-page book drawings.There are exactly two such drawings of K ; see Figure 21. We do not have a construction ofsuch drawings of K n for arbitrarily large n . Our proof of Theorem 1.1 for semisimple monotone drawings, as well as the earlier proofby ´Abrego et al. [2, Theorem 1.1], do not use all properties of monotone drawings. Bothrely only on the fact that the vertices of the drawing can be ordered as v , v , . . . , v n so thatfor every pair i, j with 1 ≤ i < j ≤ n , the vertices v i and v j are on the outer face of thedrawing induced by the interval of vertices v i , v i +1 , . . . , v j . Pedro Ramos [34] introduced theterm shellable drawings for these drawings of K n . ´Abrego et al. [3] later observed that a stillmore general condition, s -shellability for some s ≥ n/
2, is sufficient, since the depth of therecursion in the proof is only n/
2. A drawing of a complete graph with a vertex set V iscalled s -shellable if there is a subset of vertices v , v , . . . , v s ∈ V such that for every pair i, j with 1 ≤ i < j ≤ s , the vertices v i and v j are on the outer face of the drawing induced by V \ { v , v , . . . , v i − , v j +1 , v j +2 , . . . , v s } . In our version of this definition, we require v and v s to be incident with the outer face; this is slightly more restrictive compared to the originaldefinition in [3]. Informally speaking, s -shellable drawings consist of two parts: the first partis a shellable drawing of K s , the second part is an arbitrary drawing of the remaining verticesand edges that does not block the shelling of the first part. If s ≥
3, this means, in particular,that all vertices from the second part “see” the vertices in the first part in the same cyclic25rder. The class of s -shellable drawings includes, for example, all drawings with a crossing-free cycle of length s , with at least one edge of the cycle incident with the outer face [3]. Notethat the notions shellable and n -shellable coindide for drawings of K n .Following this notation, we call the sequence v , v , . . . , v n from the definition of a shellabledrawing of K n a shelling sequence of the drawing, which is similar to the term s -shelling introduced by ´Abrego et al. [3].´Abrego et al. [3] also considered the class of x -bounded drawings, which form a subclassof shellable drawings and generalize x -monotone drawings. A drawing of a graph is x -bounded if no two vertices share the same x -coordinate and every interior point of every edge uv liesin the interior of the strip bounded by two vertical lines passing through the vertices u and v . Fulek et al. [17] showed that every x -bounded drawing D can be transformed into an x -monotone drawing D ′ , while keeping the rotation system and the parity of the number ofcrossings of every pair of edges fixed. This implies, in particular, that ocr( D ) = ocr( D ′ ). Also D ′ is weakly semisimple if and only if D is weakly semisimple. Therefore, the lower boundfrom Theorem 1.1 extends to all weakly semisimple x -bounded drawings of K n .It is not a priori clear that shellable drawings are essentially different from monotone or x -bounded drawings, since the conditions for shellability and x -boundedness are very similarat first sight. In Subsection 4.2 we show that simple shellable drawings are indeed moregeneral than simple monotone drawings, but the difference is rather subtle. By a somewhatdetailed analysis, which we do not include here, it can be shown that every simple shellabledrawing of K n can be decomposed into three monotone drawings, in a very specific way.Apart from following the proof of Theorem 1.1, we may obtain a lower bound on thecrossing number of shellable drawings of K n by the following straightforward reduction to themonotone crossing number of K n , using the combinatorial characterization of x -monotonedrawings. Proposition 4.1.
Let D be a semisimple shellable drawing of K n . There is a semisimple x -monotone drawing D ′ of K n with ocr ( D ′ ) = ocr ( D ) . We note that the drawing D ′ obtained in Proposition 4.1 does not necessarily preserve theparity of the number of crossings between a given pair of edges. Moreover, it is also possiblethat for a simple shellable drawing D , we obtain a monotone drawing D ′ where some pair ofedges cross more than once; see Figure 22.Let v , v , . . . , v n be the vertices of a semisimple drawing D of K n . The order type of D is the function σ : (cid:0) [ n ]3 (cid:1) → { + , −} defined in the following way: for 1 ≤ i < j < k ≤ n , σ ( i, j, k ) = + if the triangle v i v j v k is drawn counter-clockwise and σ ( i, j, k ) = − if thetriangle v i v j v k is drawn clockwise. This generalizes the definition of the signature functionfor semisimple monotone drawings. As in the previous section, we use the shortcut σ ( i, j, k, l )for the sequence of four signs σ ( i, j, k ) σ ( i, j, l ) σ ( i, k, l ) σ ( j, k, l ). Proof.
Let v , v , . . . , v n be a shelling sequence of D . Let σ be the order type of D . We showthat σ satisfies the assumptions of Theorem 3.1, and therefore can be realized by a semisimplemonotone drawing. Let v i , v j , v k , v l be a 4-tuple of vertices with 1 ≤ i < j < k < l ≤ n .Then the drawing of K induced by v i , v j , v k , v l has v i and v l on its outer face. To verify theassumptions of Theorem 3.1, it is sufficient to show that none of the cases σ ( i, j, k, l ) = + − + ξ , σ ( i, j, k, l ) = − + − ξ , σ ( i, j, k, l ) = ξ + − + or σ ( i, j, k, l ) = ξ − + − , with ξ ∈ { + , −} , occurs.Suppose the contrary. Due to symmetry, we may suppose that σ ( i, j, k, l ) = + − + ξ . Thismeans that reading the linear counter-clockwise order of the edges incident with v i starting26
24 3 5 1 2 43 5
Figure 22: A simple shellable drawing (left) and the corresponding semisimple monotonedrawing (right). Note that the left drawing is both shellable and monotone; however, itsshelling sequence 1 , , , , v i v j before the edge v i v k , v i v k before v i v l , and v i v l before v i v j ; a contradiction.Let D ′ be a semisimple monotone drawing realizing σ . Every 4-tuple of vertices in D induces a drawing of K with at most one pair of edges crossing oddly. This is clear if D issimple; for semisimple drawings this is proved in the claim in the proof of Lemma 2.2. Calla 4-tuple of vertices in D or D ′ odd if it induces exactly one pair of edges crossing oddly and even otherwise. To finish the proof, it remains to show that odd (even) 4-tuples of verticesin D correspond to odd (even, respectively) 4-tuples in D ′ .Odd (also convex) 4-tuples in D ′ are of one of the forms ++++, −−−− , ++ −− , −− ++, − ++ − , + −− +. Even 4-tuples in D ′ are of one of the forms +++ − , − +++, −−− +, + −−− .Let v i , v j , v k , v l , with i < j < k < l , be a 4-tuple of vertices in D , inducing a drawing H of K . By deforming the plane, we may assume that v i = (0 , v l = (1 , v j , v k and the interiors of all six edges of H lie in the interior of the strip between thevertical lines passing through v i and v l . Note, however, that H is not necessarily deformableto an x -bounded drawing with v j to the left of v k : see Figure 24, left.Due to symmetry, we may assume that σ ( i, j, l ) = +. That is, the vertex v j and theinteriors of the edges v i v j and v j v l lie below the edge v i v l . Now if σ ( i, k, l ) = − , then thevertex v k and the interiors of the edges v i v k and v k v l lie above the edge v i v l . See Figure 23a). Thus, the edges v i v l and v j v k are forced to cross an odd number of times, and no otherpair of edges in H cross. Also, the triangle v i v j v k is drawn counter-clockwise and the triangle v j v k v l clockwise, so we have σ ( i, j, k, l ) = ++ −− . Therefore, the 4-tuple v i , v j , v k , v l is oddin both drawings D and D ′ .If σ ( i, k, l ) = +, then the vertex v k and the interiors of the edges v i v k and v k v l lie below theedge v i v l . We have four cases according to the signs σ ( i, j, k ) and σ ( j, k, l ), which determinethe vertical order of the edges near v i and v l , respectively, but do not determine completelywhich edges cross oddly. This is true even when the drawing H is simple; see Figure 24. If σ ( i, j, k, l ) = ++++ or σ ( i, j, k, l ) = − ++ − , then either the edges v i v k and v j v l cross oddly,or the edges v i v j and v k v l cross oddly, and some other pair of edges may cross evenly; seeFigure 23 b), c). In both cases, the 4-tuple v i , v j , v k , v l is odd in both drawings D and D ′ . If σ ( i, j, k, l ) = − +++ or σ ( i, j, k, l ) = +++ − , then no two edges cross; see Figure 23 d), e).In these last two cases, the 4-tuple v i , v j , v k , v l is even in both drawings D and D ′ .Proposition 4.1 can be generalized to weakly semisimple shellable drawings, but the equal-ity of the odd crossing numbers has to be replaced by inequality, since there are weakly27 j k l ++ −− a) b)c)d) ++++ i j k l i jk l − +++ i j k l i jk l e) +++ − i j k l − ++ − i j k l i j k li jk l Figure 23: Examples of semisimple shellable drawings of K .
31 2 4 σ (1 , , ,
4) = ++++ 21 3 4
Figure 24: Two drawings of K with the same order type.28igure 25: Left: a weakly semisimple shellable drawing of K with two pairs of edges crossingoddly. Right: a weakly semisimple drawing of K with three pairs of edges crossing oddly.semisimple shellable drawings of K with odd crossing number 2; see Figure 25, left.For general weakly semisimple drawings, the triangles are not necessarily simple closedcurves. Nevertheless, we may still define the orientation of a triangle when every two of itsedges cross evenly. Let uvw be a triangle in a weakly semisimple drawing D of K n . Orientthe closed curve γ representing the triangle uvw so that it passes through the vertices u, v, w in this cyclic order. Then for each point p on γ that is not a crossing, a sufficiently smallneighborhood of p is divided by γ into the right neighborhood and the left neighborhood of p ,consistently with the chosen orientation of γ .Let x be a point in the complement of γ in the plane. The winding number of γ around x is, informally speaking, the number of counter-clockwise turns of γ around x . More formally,if γ is parametrized by continuous polar coordinates ( r ( t ) , ϕ ( t )) : [0 , → (0 , ∞ ) × R , withcenter at x , then the winding number of γ around x is ϕ (1) − ϕ (0)2 π . We use only the parity ofthe winding number, which is independent of the chosen orientation of γ .We say that the triangle uvw , represented by the curve γ , is oriented counter-clockwise iffor some point x in the right neighborhood of u , the winding number of γ around x is even.Similarly, the triangle uvw is oriented clockwise if the winding number of γ around x is odd.Due to the fact that every two edges of uvw cross an even number of times, the definitiondoes not change if we choose x in the right neighborhood of v or w . We may thus generalizethe notion of the order type to every weakly semisimple drawing of K n with vertices labeled v , v , . . . , v n . Proposition 4.2.
Let D be a weakly semisimple shellable drawing of K n . There is a semisim-ple x -monotone drawing D ′ of K n with ocr ( D ′ ) ≤ ocr ( D ) .Proof. We proceed in the same way as in the proof of Proposition 4.1. Let v , v , . . . , v n be a shelling sequence of D and let σ be the order type of D . The fact that σ satisfiesthe assumptions of Theorem 3.1 can be proved exactly in the same way as in the proof ofProposition 4.1. Let D ′ be a semisimple monotone drawing with signature function σ .To prove the inequality, it is sufficient to show that every 4-tuple of vertices in D thatinduces a K subgraph with odd crossing number 0, corresponds to a 4-tuple with no crossingin D ′ . For that, we only need to show that the 4-tuple in D is of the type +++ − , − +++, −−− + or + −−− . All other 4-tuples in D induce subgraphs with odd crossing number 1 or2, which is at least as large as the odd crossing number of any K subgraph in D ′ .Let v i , v j , v k , v l , with i < j < k < l , be vertices in D inducing a subgraph H with all pairsof edges crossing evenly. We will show that there is a planar drawing H ′′ of the completegraph with vertices v i , v j , v k , v l , with v i and v l on its outer face, such that the orientation ofeach triangle in H ′′ is the same as in H . This will finish the proof, since such a drawing H ′′ is homeomorphic to one of the drawings in Figure 23 d), e).29he drawing H satisfies the assumptions of the weak Hanani–Tutte theorem [37]. Theweak Hanani–Tutte theorem says that for every drawing D of a graph G in the plane whereevery two edges cross an even number of times, there is a planar drawing D ′ of G which hasthe same rotation system as D (that is, the cyclic orders of the edges around each vertex arepreserved). The shortest proof of the weak Hanani–Tutte theorem, based on a more generalversion for arbitrary surfaces [33], was given by Fulek et al. [16, Lemma 3].We may assume that v i is the unique point in H with smallest x -coordinate and that v l is the unique point in H with largest x -coordinate. We extend the drawing H to a drawing K by adding a vertex y placed below H , a vertex z placed above H , and adding four edges v i y, yv l , v i z, zv l , drawn as monotone curves and forming a simple cycle v i yv l z . The cycle v i yv l z forms the boundary of the outer face of K . By the weak Hanani–Tutte theorem, thereis a planar drawing K ′ having the same rotation system as K . In particular, the cycle v i yv l z bounds a face F in K ′ . Without loss of generality, we may assume that F is the outer face of K ′ . Let H ′ be the drawing obtained from K ′ by removing the vertices y, z and their adjacentedges. Clearly, the drawings H ′ and H have the same rotation system, H ′ has no crossings,and v i and v l are on the boundary of the outer face of H ′ . The orientation of triangles v i v j v k , v i v j v l and v i v k v l is determined by the rotation at v i , and the orientation of the triangle v j v k v l is determined by the rotation at v l . It follows that H and H ′ have the same order type, andthe proof is finished.An attempt to generalize the approach in Proposition 4.1 to general non-shellable drawingsfails, for the following reason. If v , v , . . . , v n is a chosen ordering of the vertices which isnot a shelling sequence, we can have a 4-tuple v i , v j , v k , v l , with i < j < k < l , inducing aplanar drawing of K such that v i or v l is the only vertex not incident with the outer face.These 4-tuples are of type + − ++, ++ − +, − + −− , or −− + − . In monotone drawings, such4-tuples are not semisimple and, moreover, have monotone odd crossing number 2. On theother hand, this is the only obstacle in generalizing Proposition 4.1 to all simple drawings.Indeed, it is easy to see that all simple drawings of K with one crossing and arbitraryordering of the vertices are of type ++++, ++ −− , −− ++, + −− +, − ++ − , or −−−− , andthus correspond to a simple monotone drawing of K with one crossing. In fact, this is stilltrue also for semisimple drawings, by the claim in the proof of Lemma 2.2.We may thus generalize Proposition 4.1 and consequently Theorem 1.1 to every drawingof K n such that there is an ordering v , v , . . . , v n of its vertices such that for every 4-tuple v i , v j , v k , v l , with i < j < k < l , inducing a planar drawing H of K , the vertices v i and v l areon the outer face of H . We call such a drawing weakly shellable . Trivially, every drawing of K n with (cid:0) n (cid:1) crossings is weakly shellable, with arbitrary ordering of its vertices. Corollary 4.3.
Let D be a semisimple weakly shellable drawing of K n . There is a semisimple x -monotone drawing D ′ of K n with ocr ( D ′ ) = ocr ( D ) . Corollary 4.4.
Let D be a semisimple weakly shellable drawing of K n . Then ocr ( D ) ≥ Z ( n ) . We note that there are simple drawings of complete graphs that are not weakly shellable.For example, the drawing F of K in Figure 30, left, has the property that every vertex is thecentral vertex of a planar drawing of K induced by some 4-tuple of vertices. Moreover, bytaking two disjoint copies F and adding all remaining 36 edges, we obtain a simple drawingof K which will not become weakly shellable even if we change its outer face by an arbitrarysequence of edge flips. 30y removing the central vertex in F we obtain a weakly shellable simple drawing of K that is not shellable. This shows that weakly shellable drawings are more general thanshellable drawings. The definition of a shellable drawing of a complete graph involves testing a quadratic numberof subgraphs. It is easy to see that only linearly many of the subgraphs are sufficient.
Observation 4.5.
A sequence of vertices v , v , . . . , v n is a shelling sequence of a drawing ofa complete graph if and only if for every i ∈ [ n ] , the vertex v i is on the outer face of the twosubgraphs induced by the subsets of vertices { v , v , . . . , v i } and { v i , v i +1 , . . . , v n } . In a similar spirit as in Theorem 3.1, we may obtain a local characterization of shellabledrawings, by testing only the subgraphs with four vertices. Like in Theorem 3.1, we need toassume a fixed ordering of the vertices, as there are arbitrarily large minimal non-shellable(and non-monotone) drawings of complete graphs—for example, “flowers” generalizing thedrawing F in Figure 30, left. Unlike in the case of monotone drawings, the order type doesnot necessarily determine a unique shellable drawing; see Figure 24. Theorem 4.6.
Let D be a simple drawing of K n . A sequence v , v , . . . , v n of the vertices isa shelling sequence of D if and only if every -tuple v i , v j , v k , v l , with i < j < k < l , inducesa drawing of K having v i and v l on its outer face. To show Theorem 4.6, we use the following generalization of Carath´eodory’s theorem.
Lemma 4.7 (Carath´eodory’s theorem for simple complete topological graphs) . Let D be asimple drawing of K n and let x be a point in the interior of a bounded face of D . Then thereis a triangle uvw in D containing x in its interior. Moreover, there is a set of at most n − triangles covering all bounded faces of D and such that every edge of D is in at most two ofthese triangles. We use only the first part of the lemma. The stronger conclusions are included since theyfollow easily from the proof and might be interesting on their own.
Proof.
We proceed by induction on the number of vertices. For n ≤ n = 3 the statement is obvious. Now let n ≥ n − v , v , . . . , v n be the vertices of D . Let D n − be the drawing of the complete subgraph induced by v , v , . . . , v n − . Let C bethe simple curve forming the boundary of the outer face of D n − . By induction, all boundedfaces of D n − are covered by a set T n − of at most n − T n − . We assume (and prove) an even stronger inductionstatement: if two triangles from T n − share an edge e , then they do not cover the same faceincident with e . That is, the two triangles are “attached” to e from the opposite sides of e .By adding v n with its incident edges to D n − , the outer face of D n − is partitioned intothe outer face of D n and several bounded faces. We show that all these new bounded facescan be covered by a single triangle. We distinguish two cases. a) The vertex v n is in the outer face of D n − . First we observe that no edge v i v n has morethan one crossing with C . See Figure 26 a). Suppose the contrary and let x and x be two31rossings of v n v i with C closest to v n . Then the portion of v n v i between x and x separatesthe drawing D n − into two parts, each of them containing at least one vertex. In particular,the part that does not contain v i contains some other vertex v j . The edge v i v j has to lie inthe closed region bounded by C , thus it is forced to cross the edge v i v n ; a contradiction.It follows that for every edge v n v i , either the relative interior of v n v i lies outside C and v i lies on C , or v n v i crosses C in exactly one point, x i , and the portion of v n v i between x i and v i lies in the closed region bounded by C . In all cases, only the initial portion of theedge v n v i lies in the outer face of D n − . Consequently, only two edges incident with v n areincident with the outer face of D n .Let v n v k and v n v l be the two edges incident with v n and with the outer face of D n . Sincethe relative interior of the edge v k v l lies inside C , the triangle v n v k v l covers all bounded faces of D n lying outside C . If no triangle from T n − has the edge v k v l , or if exactly one such triangle, v m v k v l , exists but has the opposite orientation from v n v k v l , we let T n = T n − ∪ { v n v k v l } . Ifsome triangle v m v k v l from T n − has the edge v k v l and has the same orientation as v n v k v l ,then v m cannot lie outside v n v k v l , as then the edge v n v m would be incident with the outerface. Hence v m is inside v n v k v l . The orientation of the triangle v m v k v l then implies that thewhole triangle v m v k v l is covered by v n v k v l , and so we let T n = ( T n − \ { v m v k v l } ) ∪ { v n v k v l } . b) The vertex v n is in the interior of some bounded face of D n − . By a similar argument asin part a), every edge v n v i has at most two crossings with C . See Figure 26 b). If no edgeincident with v n is incident with the outer face of D n , then C is the boundary of the outerface of D n and thus we let T n = T n − . If two edges v n v i and v n v j cross C , they separate theclosed region bounded by C into two parts. The vertices v i and v j must be in the same part,otherwise the edge v i v j would cross v n v i or v n v j , which is forbidden.It follows that at most two edges incident with v n , v n v k and, possibly, v n v l , are incidentwith the outer face of D n . All other edges v n v i that cross C do so in a “nested fashion” in theinterval bounded by the crossings of v n v k with C , or in the interval bounded by the crossingsof v n v l with C ; see Figure 26 b). Hence, if v n v k and v n v l are incident with the outer face,then the triangle v n v k v l covers all bounded faces of D n that lie outside C .If there is no triangle v m v k v l in T n − with the same orientation as v n v k v l , we let T n = T n − ∪ { v n v k v l } . If there is a triangle v m v k v l in T n − with the same orientation as v n v k v l ,then v m has to be inside v n v k v l . For if v m was outside v n v k v l in the region bounded by v n v k , v n v l and C , then one of the edges v m v k or v m v l would be forced to cross an adjacent edge or C . Similarly, if v m was in the other region outside v n v k v l and inside (or on) C , then the edge v m v n would be forced to cross an adjacent edge or it would separate v n v k or v n v l from theouter face. Like in case a), if v m is inside v n v k v l , then the orientation of v m v k v l implies that v m v k v l is covered by v n v k v l . We let T n = ( T n − \ { v m v k v l } ) ∪ { v n v k v l } .We are left with the case when v n v k is the only edge incident with v n and with the outerface. Let x and x be the crossings of v n v k with C , so that x is between v n and x . Withoutloss of generality, assume that the portion of the edge v n v k starting at x and ending at x isoriented counter-clockwise on the boundary of the outer face. Let v n v l be the edge following v n v k clockwise in the rotation at v n .If v n v l does not cross C , then the triangle v n v k v l covers all bounded faces of D n outside C . Similarly as in the previous case, we argue that if there is a triangle v m v k v l in T n − withthe same orientation as v n v k v l , then v m is inside v n v k v l and so v m v k v l is covered by v n v k v l ,otherwise the edge v n v m would have have to cross some adjacent edge. Here we use the factthat no edge leaves v n outside the triangle v n v k v l . Again, we let T n = T n − \ { v m v k v l } ∪ n v i v j x x v n v i v j x x v n v i v j x x a) CC C v n v k v l C v n v k v l Cv m v n v k v l C b) v n v k v l C v m v n v k v l Cv m v n v k v l C v m v n v k v l Cx x x x v n v k v l C v m x x v n v k v l Cx x v m v m Figure 26: a) adding a vertex v n to the outer face. b) adding a vertex v n to a bounded face.Dotted curves represent forbidden edges. 33 v n v k v l } or T n = T n − ∪ { v n v k v l } , according to the existence of the triangle v m v k v l coveredby v n v k v l .Finally, suppose that v n v l crosses C . By induction, there is a triangle v m v i v j ∈ T n − containing v n in its interior. Hence, each of the edges v n v k and v n v l crosses at least one edgeof v m v i v j . If v n v k and v n v l cross the same edge, say, v m v i , then the edge v n v m also crosses v m v i , a contradiction. Otherwise, the region bounded by v n v k , v n v l and C that does notcontain x , contains at least one vertex of the triangle v m v i v j , say, v m . Then it is impossibleto draw the edges v m v k and v m v l so that the resulting drawing is simple. Therefore v n v l cannot cross C and we are finished. Proof of Theorem 4.6.
The condition on 4-tuples is clearly necessary. We show that it is alsosufficient. Suppose that D is a simple drawing of K n and that for some i ∈ [ n ], the vertex v i is not incident with the outer face of the subgraph induced by the subset { v , v , . . . , v i } (the case with the subset { v i , v i +1 , . . . , v n } is symmetric). By Lemma 4.7, there is a triangle v j v k v l with 1 ≤ j < k < l < i containing v i in its interior. In particular, v i is not incidentwith the outer face of the drawing of K induced by the 4-tuple v j , v k , v l , v i . Here we show that shellable drawings form a more general class than monotone drawings.We also show how monotone drawings may be characterized as a special case of shellabledrawings.Two drawings D , D of a graph G = ( V, E ) are weakly isomorphic if for every two edges e, f ∈ E , e and f cross in D if and only if they cross in D . Let D be a simple drawingof K n with vertex set { v , v , . . . , v n } . We say that a sequence of vertices v , v , . . . , v n is an x -monotone sequence of D if v and v n are incident with the outer face of D and D is weaklyisomorphic to a simple monotone drawing where v i = ( i,
0) for every i ∈ [ n ].We have the following characterization of x -monotone sequences in terms of shelling se-quences. Lemma 4.8.
Let D be a simple drawing of K n . A sequence of vertices v , v , . . . , v n is an x -monotone sequence of D if and only if it is a shelling sequence of D and the path v v . . . v n does not cross itself.Proof. The “only if” part is obvious. Let v , v , . . . , v n be a shelling sequence such that thepath v v . . . v n does not cross itself. We claim that for every v i , v j , v k , v l with 1 ≤ i < j 4, the corresponding path is noncrossing only in the left drawing.Let σ be the order type of D . By the proof of Proposition 4.1, there is a semisimplemonotone drawing D ′ with signature function σ such that two edges cross oddly in D if andonly if they cross oddly in D ′ .It remains to show that D ′ is simple. By Theorem 3.1, it is sufficient to show thatthere is no 5-tuple ( a, b, c, d, e ) with a < b < c < d < e such that σ ( a, b, e ) = σ ( a, d, e ) = σ ( b, c, d ) = σ ( a, c, e ) = ξ , where ξ ∈ { + , −} . Suppose for contrary that there is such a 5-tuple.By symmetry, we may assume that ξ = +. The vertices v a , v b , v c , v d , v e induce a shellabledrawing K of K in D . We may deform the plane by an isotopy so that v a = (0 , , v e =(1 , K are drawn between the vertical lines going through v a and v e . From σ ( a, b, e ) = + and σ ( a, c, e ) = − we have σ ( a, b, c, e ) = ++ −− . Similarly, from σ ( a, c, e ) = − and σ ( a, d, e ) = + we have σ ( a, c, d, e ) = −− ++. This further implies that σ ( a, b, c, d ) = + −− +. In particular, the edges v a v c and v b v d cross. The signatures also implythat v b and v d are below the edge v a v e and v c is above the edge v a v e . For a simple drawingthis means that the edge v b v d is below v a v e and the relative interior of the edge v a v c is above v a v e , therefore the edges v a v c and v b v d cannot cross; a contradiction.One may notice the following apparent difference between x -monotone and shellable se-quences: some drawings of K n have much more shellable sequences than x -monotone se-quences. For example, for the convex geometric drawing of K n , all n ! permutations of ver-tices are shelling sequences, whereas at most n · n − permutations of vertices, inducing anoncrossing Hamiltonian path, are x -monotone sequences.To show that shellable drawings are indeed more general than monotone drawings, weprovide an example of a shellable drawing that has no x -monotone sequence.35 a exb f Figure 28: A simple drawing S of K with shelling sequence 1 , , f, e, x, a, b, , x -monotone sequence. Theorem 4.9. The drawing in Figure 28 is a simple shellable drawing of K which is notweakly isomorphic to a simple monotone drawing.Proof. Clearly, the sequence 1 , , f, e, x, a, b, , S inFigure 28. Suppose that µ is an x -monotone sequence of S . We write v ≺ w for vertices v, w if v precedes w in µ . By symmetry, we may assume that 1 ≺ 5. The subgraphs inducedby 4-tuples { , , , } , { , , a, b } and { , , e, f } have unique x -monotone sequences, up toreversal. In particular, we have 1 ≺ ≺ ≺ 5, which in turn implies that 1 ≺ a ≺ b ≺ ≺ ≺ f ≺ e ≺ 5. To uncover the vertex a , it is not sufficient to remove the vertex 1, we haveto remove at least one more vertex. Since all vertices except for x are preceded by a in µ , wehave x ≺ a . Similarly, to uncover the vertex e , it is not sufficient to remove the vertex 5, andthe only available vertex is x . Therefore, e ≺ x . These conditions cannot be fulfilled, thus S has no x -monotone sequence. k -edges in weakly semisimple drawings Here we show a generalization of Lemma 2.2 to weakly semisimple drawings, which may beused to generalize Theorem 1.1 and the result of ´Abrego et al. [3] to weakly semisimple s -shellable drawings with s ≥ n/ 2. As in Proposition 4.2, the equality has to be replaced by aninequality. Since the orientation of triangles and hence the order type can be still defined inweakly semisimple drawings (see the definition before Proposition 4.2), the notions of k -edges, ≤ k -edges, ≤≤ k -edges and separations generalize to weakly semisimple drawings as well. Lemma 4.10. For every weakly semisimple drawing D of K n we have ocr( D ) ≥ ⌊ n/ ⌋− X k =0 E ≤≤ k ( D ) − (cid:18) n (cid:19) (cid:22) n − (cid:23) − 12 (1 + ( − n ) E ≤≤⌊ n/ ⌋− ( D ) . w w w x xv w w w Figure 29: Left: adding an auxiliary vertex and two edges to a drawing of K before applyingthe weak Hanani–Tutte theorem. Right: a planar drawing of the extended graph with thesame rotation system. Proof. The lemma follows in the same way as Lemma 2.2 or Lemma 2.1, after proving thatevery weakly semisimple drawing D of K satisfies the inequality ocr( D ) + E ( D ) ≥ 3. Theequality is not always attained as there are weakly semisimple drawings of K with oddcrossing number 3 and with six separations; see Figure 25, right.Let D be a weakly semisimple drawing of K . The separation graph of D is the subgraphof D formed by the 1-edges in D . The separation graph depends only on the order typeof D . Every order type can be obtained from each other by changing the orientation ofsome triangles. By changing the orientation of a triangle uvw , the edges uv, uw, vw changefrom 0-edges to 1-edges and vice versa. It follows that the degree of each vertex in theseparation graph either remains the same or changes by 2. Since in the planar drawing of K the separation graph is isomorphic to K , , it follows that the separation graph of D hasall vertices of odd degree. That is, it is isomorphic to K + K , K , , or K . In particular, E ( D ) ≥ D ) ≥ 1. Now suppose thatocr( D ) = 0. We show that the separation graph of D is isomorphic to K , . We achievethis by transforming D into a drawing D ′′ by a sequence of edge flips and then to a planardrawing D ′ which has the same order type as D ′′ . Performing the steps in reverse order willimply that the separation graph of each of D ′ , D ′′ and D is isomorphic to K , .Every edge flip (see the definition in the proof of Lemma 2.2) in a drawing of K changesthe orientation of two adjacent triangles. The separation graph is thus transformed by takingthe symmetric difference with a cycle C . Clearly, if the separation graph is isomorphic to K , , then its symmetric difference with arbitrarily positioned C is isomorphic to K , aswell. We may transform D by a sequence of edge flips into a drawing D ′′ which has at leastone vertex v on the outer face. Let w , w , w be the other three vertices of D ′′ , so thatthe initial portions of the edges vw and vw are incident with the outer face of D ′′ and therotation at v is w , w , w . See Figure 29, left.We extend the drawing D ′′ by adding one auxiliary vertex x close to w and edges vx and xw , so that x follows immediately after v in the rotation at w , the rotation at v is x, w , w , w , the triangle vxw is oriented clockwise and the path vxw is drawn close to theedge vw . We denote this new drawing as K .Since every two edges cross evenly in D ′′ , the same is true for the drawing K and thus wemay apply the weak Hanani–Tutte theorem to K . We obtain a planar drawing K ′ with thesame rotation system as K . We may assume that vxw w forms a boundary of the outer faceof K ′ . See Figure 29, right. Let D ′ be the subgraph of K ′ obtained after removing x and itsadjacent edges. The orientations of all three triangles incident to v are the same in D ′′ andin D ′ , since v is on the outer face in both drawings and the rotation at v is the same in K -edge2-edge3-edge4-edge Figure 30: A general simple drawing of K (left) and a cylindrical drawing of K (right)where E = 5 and E = 0, hence E ≤≤ = 10 < 12 = 3 (cid:0) (cid:1) .and in K ′ .It remains to compare the orientation of the triangle w w w in D ′′ and D ′ . Let γ ( γ ′ ) bethe closed curve formed by the edges of the triangle w w w in K ( K ′ , respectively). Sincethe curve vx crosses every edge of D ′′ an even number of times, the winding number of γ around x has the same parity as the winding number of γ around v . Since v is in the outerface of D ′′ , both winding numbers are even. Since x is outside γ ′ in K ′ , the winding numberof γ ′ around x is even as well. Together with the fact that in both drawings K and K ′ , therotation at w is the same, this implies that the triangle w w w is oriented counter-clockwisein both drawings. Therefore, D ′′ and D ′ have the same order type.Combining Lemma 4.10 with the proof by ´Abrego et al. [3], we obtain the followinggeneralization. Corollary 4.11. Let s ≥ n/ and let D be a weakly semisimple s -shellable drawing of K n .Then ocr ( D ) ≥ Z ( n ) . It would be interesting to see if techniques similar to those used in the proof of Theorem 1.1can be used to prove Hill’s conjecture for general drawings of complete graphs. We notethat the same approach does not generalize to all drawings. For example, a particular planarrealization of the so-called cylindrical drawing [21, 23] of K , with crossing number Z (10),does not satisfy the lower bound on ≤≤ K is not crossing optimal.Analogous cylindrical drawings of K k +6 , for k ≥ 2, violate the lower bound on ≤≤ k -edgesfrom Theorem 2.6.Extrapolating the definitions of ≤ k -edges and ≤≤ k -edges, we define the number of ≤≤≤ k - dges , E ≤≤≤ k ( D ), by the following identity. E ≤≤≤ k ( D ) = k X j =0 E ≤≤ j ( D ) = k X i =0 (cid:18) k + 2 − i (cid:19) E i ( D ) . In our context, using ≤≤≤ k -edges seems to be even more natural than using ≤≤ k -edges,since the formula from Lemma 2.1 can be rewritten in the following compact form:cr( D ) = 2 E ≤≤≤⌊ n/ ⌋− ( D ) − n ( n − n − 3) for n odd, andcr( D ) = E ≤≤≤⌊ n/ ⌋− ( D ) + E ≤≤≤⌊ n/ ⌋− ( D ) − n ( n − n − 2) for n even . We conjecture that the following lower bound on ≤≤≤ k -edges is satisfied by all simple draw-ings of complete graphs. Conjecture 1. Let n ≥ and let D be a simple drawing of K n . Then for every k satisfying ≤ k < n/ − , we have E ≤≤≤ k ( D ) ≥ (cid:18) k + 44 (cid:19) . Conjecture 1 is stronger than Hill’s conjecture. Theorem 2.6 implies Conjecture 1 forall simple x -monotone drawings. All our examples of simple drawings of complete graphs,including the cylindrical drawings, also satisfy Conjecture 1. We note that Conjecture 1 istrivially satisfied for k = 0, since every simple drawing of a complete graph with at least threevertices has at least three 0-edges—those incident with the outer face.We have no counterexample even to the following conjecture, which further generalizesConjecture 1 to arbitrary graphs. Conjecture 2. Let k ≥ and let D be a simple drawing of a graph with at least (cid:0) k +32 (cid:1) edges.Then E ≤≤≤ k ( D ) ≥ (cid:18) k + 44 (cid:19) . Note that in a drawing of a general graph with n vertices, a k -edge contained in t trianglesis also a ( t − k )-edge, but not necessarily an ( n − − k )-edge. Thus, for example, in everydrawing of a triangle-free graph, every edge is a 0-edge. This suggests that it might be easierto prove Conjecture 2 for non-complete graphs. Also, Conjecture 2 or some still strongervariant might be susceptible to a proof by induction on the number of edges.Further, it would be interesting to generalize Theorem 1.1 to arbitrary monotone drawings,where adjacent edges are also allowed to cross oddly. For such drawings, two notions of thecrossing number are of interest. The monotone odd crossing number , mon-ocr( G ), countingthe minimum number of pairs of edges crossing an odd number of times, and the monotoneindependent odd crossing number , mon-iocr( G ), or, mon-ocr − ( K n ), counting the number ofpairs of nonadjacent edges crossing an odd number of times. For every graph, we havemon-ocr − ( G ) ≤ mon-ocr( G ) ≤ mon-ocr ± ( G ).39 .1 Order types and λ -matrices By Lemma 2.2, the crossing number of a semisimple drawing of K n is determined by thenumber of k -edges for all k . For a set of points p , p , . . . , p n in the plane, Goodman andPollack [19] introduced the λ -matrix ( λ ( i, j )), where for every i = j , λ ( i, j ) is the number ofpoints to the left of the directed line p i p j , and λ ( i, i ) = 0. They showed that the λ -matrixdetermines the order type of the point set. Aichholzer et al. [9] used λ -matrices to representpoint sets for computing lower bounds on the rectilinear crossing number of complete graphs.The λ -matrix may be defined for semisimple drawings of K n with vertices v , v , . . . , v n in a similar way: for every i = j , λ ( i, j ) is the number of triangles v i v j v l oriented counter-clockwise. Clearly, v i v j is a k -edge if and only if λ ( i, j ) ∈ { k, n − − k } . The order type ofa drawing determines its λ -matrix, but not the drawing itself (see Figure 24 or Figure 22).Therefore, the λ -matrix does not determine the drawing either. However, a generalization ofGoodman’s and Pollack’s result to semisimple drawings is true. Observation 5.1. The λ -matrix of a semisimple drawing of K n determines its order type. This is easily seen by induction over all subgraphs of K n : in every semisimple drawing of agraph with at least one edge, all edges incident with the outer face are 0-edges. In particular,there is an edge v i v j such that λ ( i, j ) = 0. Every 0-edge determines the orientation of allincident triangles. Therefore, we may remove such an edge, update the λ -matrix and useinduction for the smaller graph.The same observation is no longer true for weakly semisimple drawings: in the drawing inFigure 25, right, every edge is a 1-edge. Therefore, its λ -matrix is identical with the λ -matrixof a mirror-symmetric drawing, but these two drawings have mutually inverse order types.Since the crossing number of a semisimple drawing of a complete graph is determined byits λ -matrix, it might be interesting to investigate the properties of λ -matrices that can berealized by semisimple drawings of complete graphs. Acknowledgments We would like to thank Pavel Valtr for initializing the research which led to this problem andMarek Eli´aˇs for developing visualization tools that were helpful during the research. References [1] B. M. ´Abrego, O. Aichholzer, S. Fern´andez-Merchant, P. Ramos and G. Salazar, The2-page crossing number of K n , Discrete Comput. Geom. (4) (2013), 747–777.[2] B. M. ´Abrego, O. Aichholzer, S. Fern´andez-Merchant, P. Ramos and G. 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