In a fan-planar drawing of a graph, each edge is either not involved in any crossing or its crossing edges have a common endpoint that is on a common side of , i.e., orienting arbitrarily and the edges towards results in a consistent orientation of the crossings on (either crosses each from left to right at each crossing, or it crosses each from right to left at each crossing); for illustrations refer to Figure 1. We call the special vertex of . All graphs in this paper are simple, that is, we do not allow parallel edges or self-loops. Hence, the vertex is uniquely defined if . If , then is an arbitrary endpoint of .
Previous literature is exclusively concerned with fan-planar drawings that are also simple, meaning that each pair of edges intersects in at most one point, which can be either an endpoint or a proper crossing. Simple drawings can be characterized in terms of two forbidden crossing configurations111In the literature, usually more obstructions are mentioned, which we exclude for all drawings (simple or not), see Section 2. (see Figure 2):
Two adjacent edges cross.
Two edges cross at least twice.
Two independent edges cross a common third edge.
Two adjacent edges cross a third edge such that their common endpoint is not on a common side of .
In this paper, we study non-simple fan-planar drawings and how to turn them into simple fan-planar drawings.
Previous and related work.
A drawing is -planar if each edge is crossed at most times and a graph is -planar, if it admits such a drawing . A -quasiplanar graph can be drawn such that no edges mutually cross – such a drawing is called -quasiplanar . Kaufmann and Ueckerdt  introduced the notion of fan-planarity in . They describe the class of graphs representable by simple fan-planar drawings222In , these graphs are called fan-planar. We do not use this terminology to avoid mix-ups with the class of graphs admitting (not necessarily simple) fan-planar drawings. as somewhere between -planar graphs and -quasiplanar graphs. Indeed, every -planar graph admits a simple -planar drawing. Since such a drawing cannot contain configuration SF1 or SF2, it is fan-planar. Moreover, a simple fan-planar drawing cannot contain three mutually crossing edges and, therefore, it is -quasiplanar. Binucci et al.  have shown that for each the class of graphs admitting simple -planar drawings and the class of graphs admitting simple fan-planar drawings are incomparable. In contrast, every so-called optimal -planar graph admits a simple fan-planar drawing . This follows from the fact that these graphs can be characterized as the graphs obtained by drawing a pentagram in the interior of each face of a pentagulation , which yields a fan-planar drawing. Angelini et al.  introduced a drawing style that combines fan-planarity with a visualization technique called edge bundling [15, 14, 20]. Each of their so-called -sided -fan-bundle-planar drawings represents a graph that is also realizable as a simple fan-planar drawing, but the converse is not true . Brandenburg  examines fan-crossing drawings, where all edges crossing a common edge share a common endpoint (in particular, this implies that SF1 is forbidden), as well as adjacency-crossing drawings, where SF1 is the only obstruction. Simple fan-planar drawings are somewhat opposite to simple -fan-crossing-free  drawings, where no adjacent edges cross another common edge.
The maximum number of edges in a simple fan-planar drawing on vertices is upperbounded by , which follows from the known density bounds for -quasiplanar graphs . A better upper bound of edges was claimed in a preprint . However, the corresponding proof appears to be flawed. We spoke with the authors and they confirmed that the current version of their proof is not correct and that they do not see a simple way to fix it333More specifically, the statement and proof of [17, Lemma 1] are incorrect. A counterexample can be obtained by removing the edge from the construction illustrated in Figure 8 (vertices correspond to the vertices in [17, Lemma 1]); for a formal description of the construction see Lemma 3. After our submission to GD’21, the authors of  have uploaded a new version  of their preprint in which they state a different definition of fan-planarity with an additional forbidden crossing configuration; also see [18, last paragraph of Section 1].. Kaufmann and Ueckerdt  described an infinite family of simple fan-planar drawings with edges. The same lower bound also follows from the aforementioned connection to optimal -planar graphs .
The recognition of graphs realizable as simple fan-planar drawings is NP-hard . The same statement also holds in the fixed rotation system setting , where the cyclic order of edges incident to each vertex is prescribed as part of the input. Consequently, efficient algorithms have only been discovered for special graph classes  and for restricted drawing styles [5, 8].
For a more comprehensive overview of previous work related to fan-planarity, we refer to a very recent survey article dedicated to fan-planarity due to Bekos and Grilli . The study of fan-planarity also falls in line with the recent trend of studying so-called beyond-planar graph classes, whose corresponding drawing styles permit crossings in restricted ways only. Apart from -planar , -quasiplanar , -fan-crossing-free , fan-bundle-planar , fan-crossing , adjacency-crossing , and fan-planar  drawings, which have already been mentioned above, several other classes of beyond-planar graphs and their corresponding drawing styles have been studied, e.g.: -gap-planar drawings  (each crossing is assigned to one of the involved edges such that each edge is assigned at most crossings), RAC-drawings  (straight-line drawings with right angle crossings), and many more. We refer to [13, 16] for recent surveys on beyond-planar graphs.
A fan-planar drawing that is not simple may contain configuration S1. Configuration S2 is allowed in a partial sense: two edges may cross any number of times, but only if orienting them arbitrarily results in a consistent orientation of their crossings; cf. Figures 1(c) and (d). Recall that every simple fan-planar drawing is -quasiplanar. In contrast, Figure 3(a) depicts a non-simple fan-planar drawing that is not -quasiplanar, which suggests that graphs admitting non-simple fan-planar drawings are not necessarily -quasiplanar. Consequently, the density bound of  for 3-quasiplanar graphs does not directly carry over. However, the depicted graph is just a , which can obviously be redrawn as a simple (fan-)planar drawing. This raises two very natural questions:
Is the largest number of edges in a -vertex non-simple fan-planar drawing larger than the number of edges in any -vertex simple fan-planar drawing?
Which non-simple fan-planar drawings can be redrawn as simple fan-planar drawings of the same graph?
Question 1 is also mentioned as an open problem by Kaufmann and Ueckerdt . Regarding question 2, we remark that the standard method for simplifying the configurations S1 and S2 does not necessarily maintain fan-planarity, see Figures 3(b) and (c). As our main result, we answer both questions, thereby solving the open problem by Kaufmann and Ueckerdt:
Every non-simple fan-planar drawing can be redrawn as a simple fan-planar drawing of the same graph without introducing additional crossings.
The proof of Theorem 1.1 is constructive and gives rise to an efficient algorithm for simplifying a given fan-planar drawing. Combined with the aforementioned previous results regarding the density [1, 17] and the recognition complexity  of graphs realizable as simple fan-planar drawings, we obtain:
Every (not necessarily simple) fan-planar drawing realizes a -quasiplanar graph.
Every (not necessarily simple) fan-planar drawing on vertices has at most edges.
Recognizing graphs that admit (not necessarily simple) fan-planar drawings is NP-hard.
In all drawings in this paper, edges are represented by simple curves. We assume no two edges touch, that is, meet tangentially. Further, we assume that no three edges share a common crossing and that edges do not contain vertices except their endpoints. Let be a drawing of a graph . A redrawing of is a drawing of . Redrawing an edge in refers to the process of obtaining a redrawing of such that .
In the beginning of Section 1, we introduced the notion of special vertices for crossed edges. To streamline the arguments, we also assign an arbitrarily chosen special vertex to each uncrossed edge. Let and be edges that cross and let be the special vertex of . We define the crossing of with as the crossing between and encountered when traversing from endpoint . For example, in Figure 5, the first crossing of with is and the second crossing is .
3 The Redrawing Procedure
We prove Theorem 1.1 by providing an algorithm that redraws the edges of a non-simple fan-planar drawing to obtain a simple fan-planar drawing. It is based on three subroutines (Lemmata 1, 2 and 4), which can be iteratively applied to remove crossings between adjacent edges (configuration S1) and multiple crossings between pairs of edges (configuration S2). More specifically, the first procedure (Lemma 1) eliminates a particular type of adjacent crossings, namely, those that involve an edge that is incident to its special vertex. The second procedure (Lemma 2) removes multiple crossings between edge pairs. Both procedures reduce the overall number of crossings. Hence, they can be exhaustively applied to obtain a redrawing of that does not contain multiple crossings between edge pairs and where adjacent crossings only involve edges that are not incident to their special vertices (Corollary 4). The procedure (Lemma 4) for removing these remaining crossings is quite involved and based on a structural analysis (Lemma 3) of the drawing .
Let be a non-simple fan-planar drawing. Let be an edge in that is incident to its special vertex . If has at least one crossing, then one of the edges in the drawing can be redrawn such that the total number of crossings in the drawing decreases. Moreover, the redrawing is fan-planar.
We continue by describing the second procedure, which eliminates crossings between pairs of edges (independent or adjacent) that cross more than once.
Let be a non-simple fan-planar drawing. Let be an edge in whose special vertex is not incident to . If edge has multiple crossings with at least one other edge, then an edge that crosses multiple times, say (where could also be incident to ), can be redrawn such that at least one crossing between and is eliminated and the total number of crossings in the drawing decreases. Moreover, the redrawing is fan-planar.
We start by describing a procedure to pick the edge that will be redrawn. We traverse from to , until the second crossing of an edge with is encountered such that the first crossing of with appeared before its second crossing, i.e., the second crossing with is closer to than its first crossing with , see Figure 5. If no such edge exists, we exchange the roles of and and repeat the procedure. We are guaranteed to find an edge with the desired properties, since there is an edge crossing multiple times.
So without loss of generality, assume that the edge has its second crossing with closer to than its first crossing . We then walk from towards along until we encounter a crossing between an edge and . The edge must also be incident to , the special vertex of .
We can now describe the redrawing procedure. The edge is redrawn to follow its previous drawing from to , cross at , follow the drawing of from to , and, finally, closely follow from to ; for an illustration see Figure 5. The following statements are proved in Appendix 0.B.
The crossing is the first crossing of with .
Redrawing maintains fan-planarity. Moreover, there is an injective mapping that assigns each crossing on the redrawn part of to a crossing on the replaced part of that involves the same edges.
Let be a non-simple fan-planar drawing. There is a fan-planar redrawing of such that
no two edges cross more than once in ;
no edge is incident to its special vertex; and
does not have more crossings than .
Adjacent crossings between edges that are not incident to their special vertices may lead to configurations where the previous edge-rerouting strategies would incur additional crossings. In the following lemma, we deal with some unproblematic cases and characterize the remaining, more challenging, configurations in terms of a sequence of conflicting edges.
Let be a non-simple fan-planar drawing in which no two edges cross more than once and such that no edge is incident to its special vertex. Let and be (adjacent) edges which cross each other at .
We can redraw such that the total number of crossings decreases and fan-planarity is maintained; or, alternatively, we can determine a sequence of edges such that the edges are pairwise distinct and the following properties are satisfied (we call the edges “red” and the edges “black”; for an illustration, see Figure 6, as well as Figure 8, which also depicts ):
is the special vertex of the black edges and incident to the red edges.
is the special vertex of the red edges and incident to the black edges.
For any odd, the first crossing of starting from is with . For any even , the first crossing of starting from is with .
crosses but no other black edge. crosses and but no other red edges.
For the purposes of the final two invariants, we define . For , let be the closed curve defined by , the arc of and the arc of , where , that connect and . For , let be the drawing induced by the edges , .
For , the curve is simple and bounds a region that contains only , an arc of that connects to and, possibly, an arc of that connects to , in its interior.
For , and is an empty triangular face in bounded by the following three arcs:
the arc of between and the special vertex of ,
the arc of between and ,
the arc of between and the special vertex of
Note that invariant 5 implies that in , crosses only and possibly . Moreover, the arcs of and connecting and via are uncrossed in .
It follows from the preconditions that is the special vertex of and is the special vertex of . We will construct the sequence of edges inductively.
We traverse from along until we encounter an edge that crosses and denote its crossing by . If and, hence, , we can redraw the part of that leads from to along such that the crossing at is removed. Moreover, since the redrawn part is crossing-free, the total number of crossings is decreased and fan-planarity is maintained. Hence, if , the statement of the lemma holds.
So assume that . It follows that, since cannot cross multiple times. Moreover, since edges are realized as simple curves. Since intersects , it is incident to .
Now, we traverse from towards until we encounter a crossing with an edge . If and, hence, , we redraw along the part of between and and the part of between and . The redrawn version of is crossing-free. Hence, we have eliminated at least one crossing (namely ) while maintaining fan-planarity and, thus, the statement of the lemma holds if .
So assume that . It follows that is distinct from since has no multiple crossings with . Moreover, since edges are simple curves. Finally, we show that . In fact, we actually claim something stronger and prove it in Appendix 0.D.
The part of between and cannot cross .
In particular, Proposition 3 implies , as claimed. Thus, we have determined two edges and such that are pairwise distinct. It remains to show that the desired invariants hold. We have already established that is incident to (since it intersects ) and, thus, 1 is satisfied for .
Since and cross , it follows that shares an endpoint with , which is the special vertex of . Accordingly, we consider two cases. First, assume that the special vertex of is , which is illustrated in Figure 7. Consider the closed curve described by , the part of between and and the part of between and . By Proposition 3 and the fact that there are no multiple crossings, the curve is indeed simple. Orient and towards . Since the resulting orientation of the crossings and has to be consistent, it follows that the part of that connects with has to intersect . More specifically, since there are no multiple crossings, it needs to intersect in some point . We now redraw along the part of that connects with and the part of that connects with . The redrawn version of only has crossings along the part between and . In particular, it crosses at , but the orientation of this crossing is consistent with the orientation of in the original drawing of . The same argument applies for all other intersected edges. Consequently, we introduce no additional crossings, eliminate the crossing , and maintain fan-planarity. Hence, the statement of the lemma holds if the special vertex of is . It remains to consider the case where the special vertex of is and, hence, is incident to . It follows that invariant 2 is satisfied for both and .
The edge cannot cross or a second time. If it crosses , then it is incident to , the special vertex of . In any case, this implies invariant 5 for .
We observe that cannot cross or since this would imply that is incident to or (the special vertex of and , respectively) and hence is parallel to or , respectively. Moreover, cannot cross a second time. Hence, the part of that leads from to is crossing-free in the drawing . Together with invariant 5 for , the invariant 5 holds for and invariant 6 holds, which concludes the base case.
Now, assume the first edges, , have been determined and . We assume all invariants hold for . Additionally, we assume that 1, 4, 5 and 6 hold for if is even (and hence is red), or 2, 4, 5 and 6 hold for if is odd (and hence is black).
. Note that in this case, we have nothing to prove for invariant 1.
If the edge has no crossings between and , then we could redraw along this part of and the arc of from to . The redrawn version of would then be uncrossed by invariant 3 for and the lemma is proved.
Otherwise has at least one crossing between and . We determine edge as follows: traverse along from towards until we encounter the first edge that crosses , let this edge be .
Invariant 3 for is satisfied by construction. To prove the remaining invariants, we establish several propositions, the proofs of which can be found in Appendix 0.E. First we prove invariant 2 for and .
Edge is incident to .
Since crosses both edges and , which are both incident to , the special vertex of is , which proves invariant 2.
Next we prove invariant 4. We first need to prove is distinct from all the previous edges.
is distinct from all edges of and does not cross edge .
The arc of between and is uncrossed in the drawing .
So the arc of between and is uncrossed in the drawing . Further, the arc of between and was uncrossed in as noted in Remark 1. Since is the only new edge introduced for , the arcs of between and as well as between and are uncrossed in . The latter in conjunction with the above proposition yields that indeed, there is a face admitting invariant 5. To see this, note that cannot cross , since it is not incident to (otherwise it would be parallel to ). Therefore, no additional edges cross while extending the subdrawing from to . Invariant 5 can be combined with the fact that the arc of from to is uncrossed by invariant 3 to conclude that the triangular region is indeed empty and invariant 6 is established.
This concludes the proof of the lemma in the case when . The second case, , is similar to the first one and can be found in Appendix 0.F. Now that we concluded the proof of Lemma 3, we have all the tools to prove Lemma 4.
Let be a non-simple fan-planar drawing with the properties established by Corollary 4. If there is an edge in that crosses an edge at and is incident to , then we can redraw an edge such that the total number of crossings in decreases, and the drawing remains fan-planar.
Let and be two adjacent edges which cross at . Their common endpoint is not the special vertex of either of the edges by Corollary 4. Thus, the special vertices of and must be and , respectively. We apply Lemma 3 on and . If can be redrawn using Lemma 3, this concludes the proof of Lemma 4. Assume that cannot be redrawn. Then we can determine a sequence of edges with the properties described in Lemma 3. We now describe how the edge can be redrawn to eliminate the crossing while maintaining fan-planarity and decreasing the overall number of crossings.
Let the other endpoint of edge be . By invariant 4, has a crossing with edge . First assume this crossing occurs between and , i.e., after enters the triangular region at . Since does not enter this region, has to leave it. It cannot cross again, nor can it cross , because it is not incident to its special vertex (note that since otherwise would be parallel to ). Finally, it cannot cross , because this is the part of that is uncrossed by invariant 3. Hence, the crossing of and cannot lie between and and must instead lie between and along . In this case, we claim that edge can be redrawn. Redraw edge to follow from until , and then follow its previous drawing from until while avoiding crossing at , as illustrated in Figure 8. We now prove that this redrawing does not introduce any new crossings on .
Redrawing does not introduce any new crossings on .
Assume a new crossing with an edge is introduced on by the redrawing operation. Since the redrawn part of is parallel to a part of , the edge crosses as well. Consequently, edge is incident to , the special vertex of .
Consider the closed curve formed by the arc of between and , the arc of between and , and the arc of between and . The edge crosses exactly once, does not cross due to invariant 4, and also does not cross since the special vertex of is due to invariant 1 and is not incident to . This implies that crosses exactly once and thus and have to lie on distinct sides of . This is illustrated in Figure 8. Edge does not cross any of the edges on the boundary of since is the only edge crossed by by Remark 1 except possibly for , and even if the part of on is still uncrossed by invariant 3, and therefore is contained in the same side of as its endpoint .
The edge crosses and is incident to , and thus must cross the curve since and lie on distinct sides of . Edge cannot be incident to since then would be parallel to . Since is the special vertex of and and is not incident to , cannot cross the edges and . Hence, must cross edge to cross the curve . Then the other endpoint of must be , the special vertex of . However, the part of connecting with is on the same side as the part of between and . Consequently, the part of connecting to and the part of between and lie on distinct sides of , which contradicts the fan-planarity. Overall, we have shown that cannot cross and, by extension, it cannot cross ; a contradiction.
The only redrawn edge is and no new crossing is introduced on , which ensures that fan-planarity is maintained. Additionally, we eliminate the crossing , which decreases the total number of crossings in the drawing.
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Appendix 0.A Proof of Lemma 1
Every edge that crosses is incident to . Traverse along the edge from to , until an edge that crosses is encountered. Let this crossing be . We redraw the edge to follow the drawing of from until and then follow its previous drawing from to without crossing at . The rerouting is illustrated in Figure 9.
The part of edge between and has no crossings by definition of . Hence, the rerouting introduces no new crossings. In particular, no new crossings are introduced on and, hence, fan-planarity is maintained. Finally, since the crossing between and at is eliminated, the total number of crossings decreases.
Appendix 0.B Proofs of the propositions in Lemma 2
Proof (of Proposition 1)
If , then we claim that , i.e., is also the first crossing of with . Assume otherwise that is the crossing of with , where . Consider the closed curve formed by the arcs of and between and . The arc of between and must cross the curve at . However, this implies that this arc of crosses itself or it crosses such that orienting towards and arbitrarily does not result in a consistent orientation of crossings. In both cases, we obtain a contradiction and, hence, is indeed the first crossing of with if .
Now assume and assume that the crossing at is the crossing of with , where . Observe that in a fan-planar drawing, if an edge has three crossings with an edge such that , where , is the crossing of with and , then appear in this order along . Consequently, the first crossing of with has to be located between and . Otherwise, when traversing from to , the first crossing of with , the second crossing of with , and the second crossing of with would be encountered in this order, which contradicts our choice of . More specifically, the first crossing of with , say , must be between and since the arc of between and has no crossings by construction; the situation is illustrated in Figure 5. Consider the closed curve again. Since is incident to and is the first crossing of with , the arc of starting from must cross between and to enter the region enclosed by before crosses at . After crosses at , it has to cross again in order to cross such that the crossing orientation of and is consistent. However, this second crossing with implies that the crossings of with or the crossings of with are not consistently oriented; a contradiction.
Hence, in any case, is the first crossing of with .
Proof (of Proposition 2)
To show that fan-planarity is maintained, we have to prove that the crossings introduced along the redrawn edge satisfy the conditions of fan-planarity. Observe that new crossings introduced along must be between and and the involved edges have to cross edge as well. Let an edge cross (and the redrawn version of ) at a point between and . If , then fan-planarity is maintained since then the crossings of the redrawing of are a subset of the crossings on its original drawing.
Assume . Let be the closed curve formed by the old drawing of between and , the arc of between and , and the arc of between and . The edge must cross by definition since lies on . Since the second crossing of with is closer to than the first crossing and due to fan-planarity, and have to lie on distinct sides of . At , we split into two parts. We use to denote the part that enters the side of that contains – the other part of is denoted by .
The special vertex of , say , must be incident to edge since crosses , and the edge must be incident to since also crosses . We distinguish two cases, namely and . We show that in both cases, crosses the part between and of the original drawing of .
First, assume . Since must be on a common side of for and , the part of that is incident to has to be . This implies that has to cross the curve by definition of . Let be the crossing of with curve that is closest to along . The crossing cannot lie on the arc of on , since otherwise and would cross as in the configuration SF2 which is forbidden. Further, cannot lie on the arc of on since this part of is uncrossed by the definition of . Hence, must lie on the old drawing of between and , i.e., along the part of formed by the old drawing of . This implies that crosses the part between and of the original drawing of .
It remains to consider the case that . Since is on a common side of for and , the part of that is incident to has to be . The arguments why crosses the part between and of the original drawing of are analogous to those used in the case .
We have shown that crosses the part between and of the original drawing of . The corresponding crossing is eliminated when redrawing , and a crossing between and is introduced after the redrawing. Hence, even though the redrawn version of crosses between and , the number of crossings does not increase. Moreover, the orientation of the crossings between and is consistent with the orientation of the crossings in the redrawn version, i.e., fan-planarity is maintained.
Appendix 0.C Proof of Corollary 4
Proof (of Corollary 4)
The redrawing procedures guaranteed by Lemma 2 and 1 decrease the number of crossings. Hence, they can be exhaustively applied to to obtain a drawing with no more crossings than such that does not satisfy the precondition of Lemma 1 or 2. In particular, if an edge is incident to its special vertex, all edges crossing it must be adjacent to it. If there is such an edge, Lemma 1 is applicable. If there is no such edge, we may choose a new special vertex for that edge, which is not incident to it. Hence, has the desired properties.
Appendix 0.D Proof of the proposition in the base case of Lemma 3
Proof (of Proposition 3)
Assume otherwise and consider the closed curve formed by the parts of and that lead from to . Both and are on the same side of since there are no multiple crossings. Since intersects , it follows that is incident to the special vertex of . Let the arc of between and be denoted by and the arc between and be denoted by . Since must lie on the same side of with respect to the two crossings and , the arc must be the arc on the side of which does not contain . However, for to be incident to , must cross , thereby crossing or a second time (recall that intersects by assumption), arriving at a contradiction.
Appendix 0.E Proof of the proposition in the inductive step of Lemma 3: Case 1
Proof (of Proposition 4)
Assume is not incident to . Since crosses and , they must have a common endpoint, which is not by assumption. Let the other endpoint of be , which implies is also incident to . Let the arc of which exits the face and enters region (which is shaded in green in Figure 10) be denoted by . To maintain fan-planarity, the vertex must be on the same side of along both edges and , which implies that the arc must end at . Endvertex is not in the region by invariant 6, hence the arc must exit the region . To exit the region , arc must cross , or . Arc cannot cross again, and also cannot cross since the arc of in this region has no crossings by invariant 3. Hence, must cross to exit the region , and must enter the region (which is shaded in blue in Figure 10). Again, endpoint is not in the region by invariant 6, and hence the arc must exit the region by crossing , or . However, cannot cross again, cannot cross since the arc of 444If this denotes again in this region is uncrossed by invariant 3, and also cannot cross since must be incident to to cross since is the special vertex of . Hence, cannot exit the region and thus cannot be incident to . We arrive at a contradiction, which implies is incident to .
Proof (of Proposition 5)
Edge is incident to , let the other endpoint be . Let the arc of which exits the face at and enters region be denoted by . To exit the region , arc would have to cross , or . Arc cannot cross again, and also cannot cross since the arc of in this region is uncrossed by invariant 3. For arc to cross , edge must be incident to , the special vertex of . However, would be parallel to in that case and the graph has no parallel edges, so this would only be possible if , but does not cross either, which implies arc cannot cross . Thus, the arc cannot exit the region . This shows that the region contains an endpoint of . Since this region was empty in by invariant 6, is distinct from all the previous edges.
Assume edge crosses edge . If the arc crosses , the crossing must be in region . This region was empty in though, so does not enter it. This also shows is not incident to or .
Therefore has to cross along the arc of between and at a crossing . We know is not incident to nor , so it doesn’t cross nor . Therefore this arc lies entirely in , since none of the arcs , and can be crossed by it. leaves when crossing at . has to cross again in order to re-enter . To cross again, has to cross , since it cannot cross itself nor again. However, if crosses at a crossing , the other end point of must be . The edge thus will enter again after crossing at , crosses at and then ends to . Consider the arc of that starts at and ends at . After crossing at , this arc enters the triangle bounded by and , see Figure 11 for an illustration. However, it is impossible for this arc to exit the triangle since none of the edges bounding the triangle can be crossed (again) by as shown above, and thus the arc of cannot be incident to , a contradiction.
Proof (of Proposition 6)
Towards a contradiction assume that the arc of between and is crossed in . Then it must be crossed by , , or , where .
Let be the crossing along the arc of between and that is closest to . is neither incident to nor , since it otherwise would be parallel to or respectively. Therefore cannot lie on any edge with these special vertices. This excludes the black edges, and . By Proposition 5, also does not cross .
Further cannot lie on the arc of between and since this arc is uncrossed by invariant 3. Thus, cannot lie on the curve , and must lie in the interior of .
Since crosses at already, cannot lie on .
Hence, must lie on a red edge where . Let be the closed curve formed by the arc of between and , the arc of between and , the arc of between and and the arc of between and . Invariant 6 applied to shows that the entirety of edge is outside . Invariant 6 applied to shows that it is completely contained in . Now let , then edge must cross since lies in , on the other side of the curve. To cross , must cross one of the arcs of the curve. cannot cross or since these arcs are uncrossed by invariant 3, and cannot cross or since it is not incident to , since invariant 2 would imply it is an edge parallel to . Thus, does not lie on a red edge.
This proves the proposition that the arc of