forcing subarrangements in complete arrangements of pseudocircles
FORCING SUBARRANGEMENTS IN COMPLETE ARRANGEMENTS OF PSEUDOCIRCLES* RONALD ORTNER Abstract. In arrangements of pseudocircles (i.e., Jordan curves) the weight of a vertex (i.e., an intersection point) is the number of pseudocircles that contain the vertex in its interior. We show that in complete arrangements (in which each two pseudocircles intersect) 2n−1 vertices of weight 0 force an α-subarrangement, a certain arrangement of three pseudocircles. Similarly, 4n − 5 vertices of weight 0 force an α4 -subarrangement (of four pseudocircles). These results on the one hand give improved bounds on the number of vertices of weight ≤ k for complete, α-free and complete, α4 -free arrangements. On the other hand, interpreting α- and α4 -arrangements as complete graphs with three and four vertices, respectively, the bounds correspond to known results in extremal graph theory.
1. Introduction An arrangement of pseudocircles is a finite set Γ = {γ1 , . . . , γn } of simple closed curves (pseudocircles) in the plane such that (i) no three curves meet each other at the same point, (ii) each two curves γi , γj have at most two points in common, and (iii) these intersection points in γi ∩ γj are always points where γi , γj cross each other. An arrangement is complete if each two pseudocircles intersect. Any arrangement can be interpreted as a planar embedding of a graph whose vertices are the intersection points between the pseudocircles and whose edges are the curves between these intersections. In the following we will often refer to this graph when talking about vertices, edges, and faces of the arrangement. Definition 1.1. Let Γ = {γ1 , . . . , γn } be an arrangement of pseudocircles. The weight of a vertex V is the number of pseudocircles γi such that V is contained in int(γi ), the interior of γi . Weights of edges and faces are defined accordingly. We will consider the number vk = vk (Γ) of vertices of given weight k, the number v≤k = v≤k (Γ) of vertices of weight ≤ k, and the number v≥k = v≥k (Γ) of vertices of weight ≥ k. Further, fk = fk (Γ) denotes the number of faces of weight k. Concerning the characterization of the weight vectors (v0 , v1 , . . . , vn−2 ) of arrangements of pseudocircles little is known. So far, sharp upper bounds on vk exist only for k = 0. Theorem 1.2 (Kedem et al. [3]). For all arrangements Γ with n := |Γ| ≥ 3, v0 ≤ 6n − 12. Date: January 2, 2014. *Some preliminary results were presented at the 24th European Workshop on Computational Geometry. 1
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Figure 1. A complete arrangement of pseudocircles with v0 = 6n − 12. Moreover, for each n ≥ 3 there is an arrangement of n (proper) circles in the plane such that v0 = 6n − 12. In a more common interpretation and originally motivated by motion planning problems [3], Theorem 1.2 shows that the complexity of the union of simple closed curves is linear in the number of elements. At its core the inductive proof of Theorem 1.2 uses that by Euler’s formula planar graphs have at most 3n − 6 edges. For arrangements with no vertices of weight > of the arrangement with vertex set V := Γ 0, the intersection graph and edge set E := {γi , γj } | γi ∩ γj 6= ∅ can be shown to be planar. As each edge in E corresponds to two vertices in the arrangement, this gives the bound of 6n − 12. More complex arrangements with vertices of weight > 0 can be disentangled so that the bound on v0 remains valid. By the circle packing theorem [4], any planar graph is the intersection graph of a circle packing and therefore (by slightly increasing the radii of the circles in the packing) also of an arrangement of (pseudo)circles. Consequently, Theorem 1.2 can be considered to be a generalization of Euler’s bound on the number of edges for planar graphs. Theorem 1.2 is sharp for complete arrangements, too. That is, for each n ≥ 3 there is a complete arrangement with v0 = 6n − 12 as shown in Figure 1. The following general upper bound on v≤k can be obtained from Theorem 1.2 by some clever probabilistic methods. Theorem 1.3 (Sharir [9]). For all arrangements of n pseudocircles and all k > 0, v≤k ≤ 26kn. We conjecture that the bound on v≤k can be improved to 6(k + 1)n. For v≥k , J. Linhart and Y. Yang established the following sharp upper bound. Theorem 1.4 (Linhart, Yang [7]). For all arrangements of n ≥ 2 pseudocircles and all k with 0 ≤ k ≤ n − 2, v≥k ≤ (n + k)(n − k − 1).
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1.1. Results. In this paper we consider a question analogous to extremal graph theory: What is the maximal v0 of a complete arrangement of n pseudocircles not containing a subarrangement of a given type? Evidently, arrangements of three pseudocircles are the smallest subarrangements of interest in this respect. Figure 2 shows the four different types one has to take into account. Subarrangements of type α play a special role here, as they are the only arrangements of
Æ
Figure 2. Complete arrangements of three pseudocircles in the plane. three pseudocircles which meet the bound of Theorem 1.2. The first result we are going to show is that 2(k + 1)n − k 2 − 3k − 1 vertices of weight k always force a subarrangement of type α. Theorem 1.5. Let Γ be a complete arrangement of n ≥ 2 pseudocircles that has no subarrangement of type α. Then v≤k ≤ 2(k + 1)n − (k + 1)(k + 2). Among all complete arrangements of four pseudocircles only one meets the bound of Theorem 1.2. In such an α4 -arrangement each subarrangement of three pseudocircles is of type α. Such arrangements prominently appear in the arrangement of Figure 1, where the three outer pseudocircles together with any other pseudocircle form an α4 -arrangement. Our main result shows that 4n − 5 vertices of weight 0 in a complete arrangement of n pseudocircles force the existence of a subarrangement of type α4 . Theorem 1.6. In complete, α4 -free arrangements of n ≥ 2 pseudocircles v0 ≤ 4n − 6. Theorems 1.5 and 1.6 correspond to well-known results of extremal graph theory. In particular, for k = 0 Theorem 1.5 gives a bound of 2n−2, which is analogous to the known bound of n − 1 on the number of edges in graphs not containing a topological K3 , the graph correspondence to an α-arrangement. Similarly, an α4 -arrangement corresponds to K4 or more precisely to a semitopological S3 , and the bound in Theorem 1.6 is analogous to the bound of 2n − 3 on the number of edges in graphs not containing a topological K4 (cf. [2]), or a semitopological S3 (cf. [10]). For further discussion see Section 5. We note that the bounds of Theorems 1.5 and Theorem 1.6 are sharp. For Theorem 1.5 this follows from the sharpness of Theorem 1.4, which is used to prove Theorem 1.5, while for Theorem 1.6 the construction in Figure 3 gives complete, α4 -free arrangements of n ≥ 2 pseudocircles that meet the bound.
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Figure 3. Complete α4 -free arrangement of six unit circles with v0 = 4n − 6. Points that look like touching points should be two intersection points between the respective circles. The same construction can be used to obtain arrangements that meet the bound for arbitrary n. 2. Forcing an α-subarrangement (Proof of Theorem 1.5) Arrangements of type α are the only complete arrangements of three pseudocircles without any face of weight 3, which is of importance in the light of the following Helly type theorem due to J. Moln´ar (see e.g. [1]). Theorem 2.1 (J. Moln´ar [8]). Let Γ = {γ1 , . . . , γn } be an arrangement of pseudocircles such that for all γi , γj , γk , int(γi ) ∩ int(γj ) ∩ int(γk ) 6= ∅. Then
Tn
i=1
int(γi ) is nonempty and simply connected.
Thus, any complete, α-free arrangement has a face of weight n. For arrangements with this property the bounds of Theorems 1.2 and Theorems 1.3 can be improved. This improvement is based on Theorem 1.4 and the following result of Y. Yang. Proposition 2.2 (Yang [11]). Let Γ be an arrangement of n pseudocircles in the plane with vertex weight vector (v0 , v1 , . . . , vn−2 ) and face weight vector (f0 , f1 , . . . , fn ) where fn > 0. Then there is an arrangement of pseudocircles Γ′ with vertex weight vector ′ (v0′ , v1′ , . . . , vn−2 ) = (vn−2 , vn−1 , . . . , v0 )
and face weight vector (f0′ , f1′ , . . . , fn′ ) = (fn , fn−1 , . . . , f0 ). Proposition 2.2 can be shown by turning the arrangement in question inside out, so that any face of weight k becomes a face of weight n − k. For a detailed proof see [6]. J. Linhart [5] pointed out that Proposition 2.2 together with Theorem 1.4 yields the following improvement of the upper bound on v≤k for arrangements with fn > 0. Theorem 2.3 (Linhart [5]). For all arrangements Γ of n ≥ 2 pseudocircles with fn > 0, v≤k ≤ 2(k + 1)n − (k + 1)(k + 2)
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for 0 ≤ k ≤ n − 2. Proof. Let Γ be an arrangement with weight vector (v0 , v1 , . . . , vn−2 ) and fn > 0. Then by Proposition 2.2, there exists an arrangement Γ′ with vk′ = vn−k−2 vertices of weight k for 0 ≤ k ≤ n − 2. Therefore, v≤k =
k X
vj =
j=0
k X
′ vn−j−2 =
j=0
n−2 X
′ vj′ = v≥n−k−2 .
j=n−k−2
Application of Theorem 1.4 then yields ′ v≤k = v≥n−k−2 ≤ (n + n − k − 2) n − (n − k − 2) − 1
=
= (2n − k − 2)(k + 1) = 2(k + 1)n − (k + 1)(k + 2).
Proof of Theorem 1.5. Since Γ has no α-subarrangement, fn (Γ) > 0, so that applying Theorem 2.3 yields the claimed bound.
3. Forcing an α4 -subarrangement (Proof of Theorem 1.6) In order to show Theorem 1.6, we start with the following result that will simplify matters, as it allows to replace the bound on v0 with a bound on f0 . Theorem 3.1. Let Γ be an arrangement of n pseudocircles. Then v0 ≤ 2n + 2f0 − 4. Theorem 3.1 will be proved without much effort from Theorem 1.2 with the aid of the subsequent upper bound on f0 in Proposition 3.2, which is also an easy consequence of Theorem 1.2. As Theorem 3.1 together with Proposition 3.2 entails Theorem 1.2, this can be considered as self-strengthening of Theorem 1.2. Proposition 3.2. Let Γ be an arrangement of n ≥ 3 pseudocircles. Then f0 ≤ 2n − 4. Proof. First note that the boundary of each bounded face of weight 0 consists of at least three edges (and hence vertices) of weight 0. For if there were a face with only two edges belonging to some pseudocircles γi and γj , then γi ∩ γj would have more than the two allowed intersection points. Concerning the unbounded face we will also assume that it has at least three edges on its boundary. In the case when it has only two edges belonging to some pseudocircles γi and γj , then all other pseudocircles are contained in int(γi ) ∪ int(γj ) so that f0 = 1 and the bound trivially holds. On the other hand, each vertex of weight 0 is on the boundary of only a single face of weight 0. Therefore by Theorem 1.2, f0 ≤
6n − 12 v0 ≤ = 2n − 4. 3 3
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Figure 4. Adding a pseudocircle γ that intersects each edge of a face in two vertices of weight 0. For each new face of weight 0 there are two new vertices of weight 0. In order to enhance readability, in the following we will use the term 0-face to refer to faces of weight 0. Further, for a face F in an arrangement ∂F shall denote the boundary of F . Proof of Theorem 3.1. We start showing the theorem for two particular cases. First, if the unbounded face has only two edges on its boundary, then as argued in the proof of Proposition 3.2 we have f0 = 1 as well as v0 = 2 and n ≥ 2, so that the theorem holds. Second, let us consider the case when there is a 0-face F with more than three edges such that only three pseudocircles γi , γj , γk ∈ Γ contribute edges to ∂F . Then the same face F will appear in the subarrangement {γi , γj , γk }. As can be seen from the classification of complete arrangements of three pseudocircles in Figure 2, F must be the unbounded face of a β-arrangement, as the 0-faces of all other arrangements have at most three edges. Consequently, F is the only face of weight 0 in Γ and we have f0 = 1, v0 = 4 as well as n ≥ 3, so that the theorem holds. Thus, let us now assume that the unbounded 0-face and therefore all 0-faces have at least three edges on its boundary and that all 0-faces with more than three edges have more than three pseudocircles contributing to its boundary. Let f0,i be the number of 0faces with i edges. We prove the theorem by induction on the tuples (f0,n , f0,n−1 , . . . , f0,4 ) in lexicographical order. First, if all f0,i with i > 3 are 0, then all 0-faces are triangles, so that v0 = 3f0 and by Proposition 3.2, v0 = 2f0 + f0 ≤ 2f0 + 2n − 4. If there is at least one 0-face F with more than three edges, we may add a pseudocircle γ which intersects for each pseudocircle contributing to ∂F one boundary edge of F (cf. Figure 4, which shows the case when each pseudocircle contributes one edge to ∂F ). By assumption, more than three pseudocircles contribute edges to F , so that in the resulting arrangement Γ′ there is an f0,i (i > 3) being smaller than in Γ, while only f0,j with i > j may have increased. Therefore, the induction assumption is applicable. Now, if f0 has increased by ℓ, then v0 has increased by 2(ℓ + 1), so that by induction assumption v0 (Γ) + 2(ℓ + 1) = v0 (Γ′ ) ≤ 2(n + 1) + 2 f0 (Γ) + ℓ − 4, whence the theorem follows.
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Theorem 1.6 now is an immediate consequence of the following bound on f0 (for a proof we refer to Section 4) together with Theorem 3.1. Theorem 3.3. In complete, α4 -free arrangements of n ≥ 2 pseudocircles, f0 ≤ n − 1. 4. Proof of Theorem 3.3 4.1. Preliminaries. For α-arrangements Γα we denote the bounded 0-face of Γα by Fˆ0 (Γα ). Sloppily, we will often say that a point or face is inside Γα , meaning that it is contained in Fˆ0 (Γα ). Correspondingly, by saying that something is outside Γα , we mean that it is contained in the unbounded 0-face of Γα . We start with some simple facts about the topology of (complete) arrangements. The aim is to establish that in each complete arrangement with more than one 0-face there is an α-arrangement Γα , such that all bounded 0-faces are contained in Fˆ0 (Γα ). Observation 4.1. Let Γ be an arrangement of pseudocircles and γ ∈ Γ. Then two 0faces F1 , F2 of Γ are merged into one when removing γ if and only if γ has an edge ei of weight 0 on ∂Fi (i = 1, 2) and e1 and e2 can be connected by a curve only contained in the interior of γ (and especially not intersecting any other pseudocircle). In particular, such 0-faces F1 , F2 exist when f0 (Γ) > f0 (Γ \ {γ}). In the following, we will say that γ separates F or that γ separates F1 from F2 , if by removing γ from Γ two 0-faces F1 , F2 are merged into a 0-face F in Γ \ {γ} with F1 , F2 ⊆ F . Proposition 4.2. Let Γ be a complete arrangement of pseudocircles with f0 (Γ) > 1. Then Γ has an α-subarrangement. Proof. We show that for α-free arrangements f0 = 1. Assume that there is an α-free arrangement Γ with f0 (Γ) > 1. By Theorem 2.1, fn (Γ) = 1, so that we may apply Proposition 2.2 to conclude that there is an arrangement Γ′ with fn (Γ′ ) > 1. But this contradicts Theorem 2.1. Lemma 4.3 (Separation Lemma). Let F + , F − be two 0-faces in some complete arrangement Γ of pseudocircles. Then there is an α-arrangement Γα ⊆ Γ such that F + is inside Γα , while F − is outside Γα (or vice versa). Proof. First remove pseudocircles from Γ one after another, such that F + and F − are not merged. When this is no longer possible, then in the resulting arrangement Γ′ = {γ1 , . . . , γk } each remaining γi separates F + from F − . That is, on the boundary of each − + + − γi there are two 0-edges e+ and e− i and ei such that ei is on ∂F i is on ∂F . By Proposition 4.2, Γ′ contains some α-arrangement Γα = {γi1 , γi2 , γi3 }. Assume that both − + − + − F + and F − are contained in the same 0-face of Γα . Then the edges e+ i1 , ei1 , ei2 , ei2 , ei3 , ei3 are all contained on the same boundary curve C of Γα . However, it is not possible to − + distribute the edges e+ j , ej on the pseudocircles γj (j = i1 , i2 , i3 ) such that the ej can be
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e+ i
γi
e+ j
e− i Fˆ0 (Γα )
F−
F+
γj e− j
Figure 5. Illustration for the proof of Lemma 4.4 when |Γ∗ | ≥ 2. + separated from the e− and F − cannot be contained in the same j on C. It follows that F 0-face of Γα .
Lemma 4.4. Let F + , F − be two 0-faces in some complete arrangement Γ that are both inside some α-arrangement Γα ⊆ Γ. Then there is another α-arrangement Γ+ α ⊆ Γ with + + − + F inside Γα , and F outside Γα , such that all 0-faces of Γ that are inside Γ+ α are also inside Γα . Proof. By the Separation Lemma 4.3 there is an α-arrangement Γ′α that separates F + from F − . Remove pseudocircles from Γ′α as long as the faces F + , F − remain separated, and let Γ∗ be a minimal subset of Γ′α such that when removing a pseudocircle in Γ∗ from Γ∗ ∪ Γα the faces F + , F − are merged. (Here and in the following F + and F − actually denote the possibly larger 0-faces in Γ∗ ∪ Γα containing the original faces F + and F − .) If Γ∗ contains only a single element γ then F + and F − are separated in Γα ∪ {γ}, and it is + easy to see that γ forms with suitable pseudocircles in Γα an α-arrangement Γ+ α with F ˆ inside and F − outside. Furthermore, Fˆ0 (Γ+ α ) ⊆ F0 (Γα ) so that all 0-faces of Γ that are inside Γ+ α are also inside Γα . Now assume that |Γ∗ | ≥ 2. As no pseudocircle ∈ Γ∗ can be removed without merging − F + and F − , for each pseudocircle γi ∈ Γ∗ there are two 0-edges e+ i , ei on the boundary − − + + − of γi such that e+ i is on ∂F , ei is on ∂F , and ei can be connected with ei by a + − curve only contained in int(γi ). Now, ei and ei must lie on the same component Ci of + γi ∩ Fˆ0 (Γα ): Otherwise either e− i cannot be connected with ei by a curve only contained in int(γi ), or F + and F − are separated in Γα ∪ {γi }, so that one could choose Γ∗ = {γi }. − Furthermore, e+ i and ei obviously cannot be neighbors on Ci , so that there must be a − γj ∈ Γ∗ that separates e+ i from ei on Ci . Thus, by completeness of the arrangement, the situation is as shown in Figure 5. Note that by Observation 4.1, γj ∩ ∂ Fˆ0 (Γα ) cannot lie − in the interior of γi , as in this case it would be impossible to connect the edges e+ i , ei by a curve only contained in the interior of γi . Similarly, γi ∩ ∂ Fˆ0 (Γα ) cannot lie in the interior of γj . Thus, for each γ ∈ Γα it holds that neither int(γ) contains a point from γi ∩ γj , nor can either int(γi ) or int(γj ) contain a point of γ ∩ γj or γ ∩ γi , respectively. It follows that γi , γj together with any γ ∈ Γα forms an α-arrangement Γγ such that all 0-faces of Γ that are inside Γγ are also inside Γα . Moreover, it is easy to see that for a suitable γ the 0-face F + is inside Γγ while F − is outside Γγ . Repeated application of Lemma 4.4 gives the following result.
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Corollary 4.5. Let Γ be a complete arrangement with bounded 0-face F . Then there is an α-subarrangement Γα in Γ such that F is the only 0-face inside Γα . Projecting the arrangement to the sphere the unbounded face of weight 0 becomes bounded, while Corollary 4.5 is still applicable, which therefore yields the following counterpart to Corollary 4.5. Corollary 4.6. Given a complete arrangement Γ with f0 (Γ) > 1, there is an α-subarrangement Γα in Γ such that all bounded 0-faces are inside Γα .
4.2. Proof of Theorem 3.3. We conclude with the following lemma from which Theorem 3.3 follows immediately. Lemma 4.7. Let Γ be a complete, α4 -free arrangement with f0 (Γ) > 1, and let Γα be an α-subarrangement of Γ that contains all bounded 0-faces in its interior (cf. Corollary 4.6). Then there is a γ ∈ Γα such that removing γ from Γ loses at most one face of weight 0, i.e., f0 (Γ \ {γ}) ≥ f0 (Γ) − 1. Proof of Theorem 3.3. The theorem is trivial for n = 2, while for n = 3 it can be verified from Figure 2. We proceed by induction on n. If f0 (Γ) ≤ 1 the theorem again holds trivially for n ≥ 2. If f0 (Γ) > 1 then Lemma 4.7 allows to remove a pseudocircle γ, so that f0 is decreased by at most 1. Together with the induction assumption for Γ \ {γ} this shows the theorem. Proof of Lemma 4.7. We derive a contradiction from the assumption that for each γi ∈ Γα = {γ1 , γ2, γ3 }, f0 (Γ \ {γi }) < f0 (Γ) − 1 (i.e., removing any pseudocircle in Γα from Γ decreases f0 by at least 2). Under this assumption, each γi ∈ Γα borders on at least three 0-faces, two of which (by assumption on Γα ) are contained in Fˆ0 (Γα ). According to Observation 4.1 these two faces Fi,1 , Fi,2 can be connected by a curve Ci only contained in the interior of γi . Each Ci separates the interior of γi into two regions Ri− , Ri+ , where we assume Ri+ to be the region that borders on the unbounded face of weight 0, see Figure 6. On the other hand, Fi,1 , Fi,2 must be separated by some other pseudocircle: For each pseudocircle γi in Γα there is a pseudocircle γi′ ∈ / Γα that intersects γi in Ri− (i = 1, 2, 3). Any such γi′ separates Fi,1 from Fi,2 . Note that both {γi , γi′ , γj } (j 6= i) and {γi , γi′ , γk } (k 6= i, j) are α-arrangements, since γi′ cannot intersect γj (j 6= i) in the interior of γi without crossing Ci (which cannot happen according to Observation 4.1). Summarizing, one of the two 0-faces Fi,1 , Fi,2 is contained inside an α-arrangement {γi , γi′ , γj }, the other contained inside an α-arrangement {γi , γi′ , γk }, cf. Figure 6.
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+ + + + + + + . . . . + . . . . . + . . . .. . . .. . + + . . . + . . . . . . . + + . +
γ1
+
γ1′
+ +
γ3′
+
+
+ + +
+ +
γ2
+.
γ2′
+ + + + + + + . . . . .. . . . . + .. ... . . .. . . + . . .. . + . . . .. + . + + + +
. + ... . . . . .. .. . ... . . .. . . . . . . + + . .. .. . . . . . + + . . . . + + +
γ3
+
Figure 6. Illustration of the situation in Lemma 4.7. The curves Ci separate int(γi ) into regions Ri+ and Ri− (dotted). The pseudocircles γi′ intersect γi in Ri− . + + + + + + + . . . . + . . . . . + . . . .. . . .. . + + . . . + . . . . . . . + + . +
γ1
+
+
γj′
+ +
+
+ + +
+
+.
+ +
γ2
+ + + + + + + . . . . .. . . . . + . .. .. . . .. . . + . . .. . + . . . .. + . + + + +
. + ... . . . . .. .. . ... . . .. . . . . . . + . . . . . . . . + . . . . . . + + + +
γ3
+
Figure 7. If γj′ intersects Ri− , γ3 can be intersected only by allowing an α4 -arrangement. We distinguish the following two cases: ⊲ |{γ1′ , γ2′ , γ3′ }| = 1: In this case γ1′ = γ2′ = γ3′ , so that Γα ∪ {γ1′ } is an α4 -arrangement, contrary to our assumption. ⊲ |{γ1′ , γ2′ , γ3′ }| ≥ 2: Assume without loss of generality that γ1′ 6= γ2′ . Let us consider the intersections of γj′ with γi for i, j ∈ {1, 2} and i 6= j. First note that int(γj′ ) cannot intersect both Ri+ and Ri− , as this would cause a forbidden intersection with Ci . Now, if γj′ intersects Ri− , γj′ cannot intersect γ3 in the interior of either γi or γj without intersecting either Ci or Cj . Therefore, in this case {γi , γj , γ3 , γj′ } is a forbidden α4 -arrangement, cf. Figure 7. Thus let us assume that γ2′ intersects R1+ and γ1′ intersects R2+ , and consequently int(γ2′ ) ∩ R1− = ∅ as well as int(γ1′ ) ∩ R2− = ∅. Since we also have int(γi′ ) ∩ Ri+ = ∅ for i = 1, 2, it follows that int(γ1′ ) ∩ int(γ2′ ) ∩ int(γ1 ) = ∅ and int(γ1′ ) ∩ int(γ2′ ) ∩ int(γ2 ) = ∅. Therefore {γ1′ , γ2′ , γ1} and {γ1′ , γ2′ , γ2 } are α-arrangements. We have already argued above that {γ1 , γ1′ , γ2} and {γ2 , γ2′ , γ1 } are α-arrangements as well, so that {γ1 , γ2, γ1′ , γ2′ } is a forbidden α4 -arrangement.
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5. Discussion and Open Problems 5.1. Consequences. The improved upper bound on v0 of Theorem 1.6 can in turn be used to improve the upper bound on v≤k for complete, α4 -free arrangements. Theorem 5.1. For complete, α4 -free arrangements of n ≥ 2 pseudocircles and k > 0, v≤k ≤ 18kn. Proof. The proof is basically identical to the proof of Theorem 1.3 in [9], only with the application of Theorem 1.2 replaced by an application of Theorem 1.6 and the constants adapted accordingly. Remark 5.2. As α4 -arrangements cannot be realized with unit circles, Theorems 1.6 and 5.1 hold in particular for complete arrangements of unit circles. Note that for general arrangements of unit circles no significant improvement of Theorem 1.2 is possible: Consider the arrangement obtained from a densest circle packing by increasing the radius of the circles by a sufficiently small value, so that each touching point is transformed into two intersection points of weight 0. Then each circle intersects six others, which gives a total of ≈ 6n vertices of weight 0. 5.2. Generalization. A natural question is whether the results of Theorems 1.5 and 1.6 can be generalized to arbitrary (i.e., not necessarily complete) arrangements. We conjecture that a pendant to Theorem 1.5 holds for arrangements that do not contain a cyclic chain according to the following definition. Definition 5.3. A cyclic chain is an arrangement Γ whose elements can be indexed such that Γ = {γ1 , . . . , γn }, and γi intersects γi+1 in two vertices of weight 0 for all i = 1, . . . , n (with γn+1 := γ1 ). Also, there are no further intersections. Our current proof of Theorem 1.5 cannot be easily adapted to show this conjecture. Still, if one is able to generalize the result of Theorem 1.4 to certain arrangements on the sphere (it is easy to see that Theorem 1.4 cannot hold for all arrangements on the sphere), a modification may succeed. Concerning Theorem 1.6, it is not even clear what a generalization may look like. An obvious generalization of an α4 -arrangement would be that of a wheel. Definition 5.4. A wheel is an arrangement which consists of a cyclic chain Γ′ plus another pseudocircle γ which separates one of the two faces of weight 0 in Γ′ into at least three faces of weight 0. Indeed, for arrangements in which all vertices have weight 0, the following result can be considered as a generalization of Theorem 1.6. It follows immediately by application of the following, already mentioned result of C. Thomassen [10] to the arrangement’s intersection graph (as defined after Theorem 1.2): Any graph G = (V, E) that does not contain a semitopological graph S3 (i.e., a cycle plus an additional vertex that is connected to three vertices in the cycle) has at most 2|V | − 3 edges.
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Figure 8. Difficulties when trying to generalize Theorem 1.6. Proposition 5.5. For wheel-free arrangements of n ≥ 2 pseudocircles with no vertices of weight > 0, v0 ≤ 4n − 6. However, Proposition 5.5 does not hold in general, as the arrangement on the left of Figure 8 shows. Although the arrangement does not contain a wheel,1 neither of the bounds f0 ≤ n − 1 and v0 ≤ 4n − 6 holds. On the other hand, simply allowing further intersection points in the definition of cyclic chain is too generous. While it is possible to obtain a variant of Theorem 1.6 for arbitrary arrangements, this result is not a proper generalization of Theorem 1.6: The (complete) arrangement on the right of Figure 8 is α4 -free, but not wheel-free according to the modified definition. Thus, it is not clear what a good generalization of Theorem 1.6 may look like. Acknowledgments. The author would like to thank J. Linhart for helpful comments and suggestions on prior versions of this paper as well as G¨ unter Rote and two anonymous reviewers for pointing out some errors and other suggestions that led to a considerably shorter proof of Lemma 4.7 and general improvement of the paper. A revision of the paper has been prepared while the author was supported by the Austrian Science Fund (FWF): J 3259-N13. References [1] M. Breen, A Helly-type theorem for intersections of compact connected sets in the plane, Geom. Dedicata 71 (1998), 111–117. [2] G. A. Dirac, In abstrakten Graphen vorhandene vollst¨andige 4-Graphen und ihre Unterteilungen, Math. Nachr. 22 (1960), 61–85. [3] K. Kedem, R. Livne, J. Pach, and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1 (1986), 59–71. [4] P. Koebe, Paul , Kontaktprobleme der konformen Abbildung, Ber. Verh. S¨ achs. Akad. Leipzig, Math.-Phys. Kl. 88 (1936), 141–164 . [5] J. Linhart, private communication. [6] J. Linhart, R. Ortner, A note on convex realizability of arrangements of pseudocircles. Geombinatorics XVIII/2 (2008), 66–71. 1As
the two circles intersect (inside the ellipse), the pseudocircles outside do not form a cyclic chain as defined in Definition 5.3.
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[7] J. Linhart, Y. Yang, Arrangements of arcs and pseudocircles, Beitr¨ age Algebra Geom. 37/2 (1996), 391–398. ¨ [8] J. Moln´ ar, Uber den zweidimensionalen topologischen Satz von Helly (in Hungarian, Russian and German summary), Mat. Lapok 8 (1957), 108–114. [9] M. Sharir, On k-sets in arrangements of curves and surfaces, Discrete Comput. Geom. 6 (1991), 593–613. [10] C. Thomassen A minimal condition implying a special K4 -subdivision in a graph, J. Arch. Math. (Basel) 25 (1974), 210–215. [11] Y. Yang, Arrangements of circles on E 2 , unpublished typescript.
E-mail address: [email protected] ¨t Leoben Department Mathematik und Informationstechnologie, Montanuniversita Franz-Josef-Strasse 18, A-8700 Leoben, Austria
1. Introduction An arrangement of pseudocircles is a finite set Γ = {γ1 , . . . , γn } of simple closed curves (pseudocircles) in the plane such that (i) no three curves meet each other at the same point, (ii) each two curves γi , γj have at most two points in common, and (iii) these intersection points in γi ∩ γj are always points where γi , γj cross each other. An arrangement is complete if each two pseudocircles intersect. Any arrangement can be interpreted as a planar embedding of a graph whose vertices are the intersection points between the pseudocircles and whose edges are the curves between these intersections. In the following we will often refer to this graph when talking about vertices, edges, and faces of the arrangement. Definition 1.1. Let Γ = {γ1 , . . . , γn } be an arrangement of pseudocircles. The weight of a vertex V is the number of pseudocircles γi such that V is contained in int(γi ), the interior of γi . Weights of edges and faces are defined accordingly. We will consider the number vk = vk (Γ) of vertices of given weight k, the number v≤k = v≤k (Γ) of vertices of weight ≤ k, and the number v≥k = v≥k (Γ) of vertices of weight ≥ k. Further, fk = fk (Γ) denotes the number of faces of weight k. Concerning the characterization of the weight vectors (v0 , v1 , . . . , vn−2 ) of arrangements of pseudocircles little is known. So far, sharp upper bounds on vk exist only for k = 0. Theorem 1.2 (Kedem et al. [3]). For all arrangements Γ with n := |Γ| ≥ 3, v0 ≤ 6n − 12. Date: January 2, 2014. *Some preliminary results were presented at the 24th European Workshop on Computational Geometry. 1
2
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Figure 1. A complete arrangement of pseudocircles with v0 = 6n − 12. Moreover, for each n ≥ 3 there is an arrangement of n (proper) circles in the plane such that v0 = 6n − 12. In a more common interpretation and originally motivated by motion planning problems [3], Theorem 1.2 shows that the complexity of the union of simple closed curves is linear in the number of elements. At its core the inductive proof of Theorem 1.2 uses that by Euler’s formula planar graphs have at most 3n − 6 edges. For arrangements with no vertices of weight > of the arrangement with vertex set V := Γ 0, the intersection graph and edge set E := {γi , γj } | γi ∩ γj 6= ∅ can be shown to be planar. As each edge in E corresponds to two vertices in the arrangement, this gives the bound of 6n − 12. More complex arrangements with vertices of weight > 0 can be disentangled so that the bound on v0 remains valid. By the circle packing theorem [4], any planar graph is the intersection graph of a circle packing and therefore (by slightly increasing the radii of the circles in the packing) also of an arrangement of (pseudo)circles. Consequently, Theorem 1.2 can be considered to be a generalization of Euler’s bound on the number of edges for planar graphs. Theorem 1.2 is sharp for complete arrangements, too. That is, for each n ≥ 3 there is a complete arrangement with v0 = 6n − 12 as shown in Figure 1. The following general upper bound on v≤k can be obtained from Theorem 1.2 by some clever probabilistic methods. Theorem 1.3 (Sharir [9]). For all arrangements of n pseudocircles and all k > 0, v≤k ≤ 26kn. We conjecture that the bound on v≤k can be improved to 6(k + 1)n. For v≥k , J. Linhart and Y. Yang established the following sharp upper bound. Theorem 1.4 (Linhart, Yang [7]). For all arrangements of n ≥ 2 pseudocircles and all k with 0 ≤ k ≤ n − 2, v≥k ≤ (n + k)(n − k − 1).
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1.1. Results. In this paper we consider a question analogous to extremal graph theory: What is the maximal v0 of a complete arrangement of n pseudocircles not containing a subarrangement of a given type? Evidently, arrangements of three pseudocircles are the smallest subarrangements of interest in this respect. Figure 2 shows the four different types one has to take into account. Subarrangements of type α play a special role here, as they are the only arrangements of
Æ
Figure 2. Complete arrangements of three pseudocircles in the plane. three pseudocircles which meet the bound of Theorem 1.2. The first result we are going to show is that 2(k + 1)n − k 2 − 3k − 1 vertices of weight k always force a subarrangement of type α. Theorem 1.5. Let Γ be a complete arrangement of n ≥ 2 pseudocircles that has no subarrangement of type α. Then v≤k ≤ 2(k + 1)n − (k + 1)(k + 2). Among all complete arrangements of four pseudocircles only one meets the bound of Theorem 1.2. In such an α4 -arrangement each subarrangement of three pseudocircles is of type α. Such arrangements prominently appear in the arrangement of Figure 1, where the three outer pseudocircles together with any other pseudocircle form an α4 -arrangement. Our main result shows that 4n − 5 vertices of weight 0 in a complete arrangement of n pseudocircles force the existence of a subarrangement of type α4 . Theorem 1.6. In complete, α4 -free arrangements of n ≥ 2 pseudocircles v0 ≤ 4n − 6. Theorems 1.5 and 1.6 correspond to well-known results of extremal graph theory. In particular, for k = 0 Theorem 1.5 gives a bound of 2n−2, which is analogous to the known bound of n − 1 on the number of edges in graphs not containing a topological K3 , the graph correspondence to an α-arrangement. Similarly, an α4 -arrangement corresponds to K4 or more precisely to a semitopological S3 , and the bound in Theorem 1.6 is analogous to the bound of 2n − 3 on the number of edges in graphs not containing a topological K4 (cf. [2]), or a semitopological S3 (cf. [10]). For further discussion see Section 5. We note that the bounds of Theorems 1.5 and Theorem 1.6 are sharp. For Theorem 1.5 this follows from the sharpness of Theorem 1.4, which is used to prove Theorem 1.5, while for Theorem 1.6 the construction in Figure 3 gives complete, α4 -free arrangements of n ≥ 2 pseudocircles that meet the bound.
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Figure 3. Complete α4 -free arrangement of six unit circles with v0 = 4n − 6. Points that look like touching points should be two intersection points between the respective circles. The same construction can be used to obtain arrangements that meet the bound for arbitrary n. 2. Forcing an α-subarrangement (Proof of Theorem 1.5) Arrangements of type α are the only complete arrangements of three pseudocircles without any face of weight 3, which is of importance in the light of the following Helly type theorem due to J. Moln´ar (see e.g. [1]). Theorem 2.1 (J. Moln´ar [8]). Let Γ = {γ1 , . . . , γn } be an arrangement of pseudocircles such that for all γi , γj , γk , int(γi ) ∩ int(γj ) ∩ int(γk ) 6= ∅. Then
Tn
i=1
int(γi ) is nonempty and simply connected.
Thus, any complete, α-free arrangement has a face of weight n. For arrangements with this property the bounds of Theorems 1.2 and Theorems 1.3 can be improved. This improvement is based on Theorem 1.4 and the following result of Y. Yang. Proposition 2.2 (Yang [11]). Let Γ be an arrangement of n pseudocircles in the plane with vertex weight vector (v0 , v1 , . . . , vn−2 ) and face weight vector (f0 , f1 , . . . , fn ) where fn > 0. Then there is an arrangement of pseudocircles Γ′ with vertex weight vector ′ (v0′ , v1′ , . . . , vn−2 ) = (vn−2 , vn−1 , . . . , v0 )
and face weight vector (f0′ , f1′ , . . . , fn′ ) = (fn , fn−1 , . . . , f0 ). Proposition 2.2 can be shown by turning the arrangement in question inside out, so that any face of weight k becomes a face of weight n − k. For a detailed proof see [6]. J. Linhart [5] pointed out that Proposition 2.2 together with Theorem 1.4 yields the following improvement of the upper bound on v≤k for arrangements with fn > 0. Theorem 2.3 (Linhart [5]). For all arrangements Γ of n ≥ 2 pseudocircles with fn > 0, v≤k ≤ 2(k + 1)n − (k + 1)(k + 2)
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for 0 ≤ k ≤ n − 2. Proof. Let Γ be an arrangement with weight vector (v0 , v1 , . . . , vn−2 ) and fn > 0. Then by Proposition 2.2, there exists an arrangement Γ′ with vk′ = vn−k−2 vertices of weight k for 0 ≤ k ≤ n − 2. Therefore, v≤k =
k X
vj =
j=0
k X
′ vn−j−2 =
j=0
n−2 X
′ vj′ = v≥n−k−2 .
j=n−k−2
Application of Theorem 1.4 then yields ′ v≤k = v≥n−k−2 ≤ (n + n − k − 2) n − (n − k − 2) − 1
=
= (2n − k − 2)(k + 1) = 2(k + 1)n − (k + 1)(k + 2).
Proof of Theorem 1.5. Since Γ has no α-subarrangement, fn (Γ) > 0, so that applying Theorem 2.3 yields the claimed bound.
3. Forcing an α4 -subarrangement (Proof of Theorem 1.6) In order to show Theorem 1.6, we start with the following result that will simplify matters, as it allows to replace the bound on v0 with a bound on f0 . Theorem 3.1. Let Γ be an arrangement of n pseudocircles. Then v0 ≤ 2n + 2f0 − 4. Theorem 3.1 will be proved without much effort from Theorem 1.2 with the aid of the subsequent upper bound on f0 in Proposition 3.2, which is also an easy consequence of Theorem 1.2. As Theorem 3.1 together with Proposition 3.2 entails Theorem 1.2, this can be considered as self-strengthening of Theorem 1.2. Proposition 3.2. Let Γ be an arrangement of n ≥ 3 pseudocircles. Then f0 ≤ 2n − 4. Proof. First note that the boundary of each bounded face of weight 0 consists of at least three edges (and hence vertices) of weight 0. For if there were a face with only two edges belonging to some pseudocircles γi and γj , then γi ∩ γj would have more than the two allowed intersection points. Concerning the unbounded face we will also assume that it has at least three edges on its boundary. In the case when it has only two edges belonging to some pseudocircles γi and γj , then all other pseudocircles are contained in int(γi ) ∪ int(γj ) so that f0 = 1 and the bound trivially holds. On the other hand, each vertex of weight 0 is on the boundary of only a single face of weight 0. Therefore by Theorem 1.2, f0 ≤
6n − 12 v0 ≤ = 2n − 4. 3 3
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Figure 4. Adding a pseudocircle γ that intersects each edge of a face in two vertices of weight 0. For each new face of weight 0 there are two new vertices of weight 0. In order to enhance readability, in the following we will use the term 0-face to refer to faces of weight 0. Further, for a face F in an arrangement ∂F shall denote the boundary of F . Proof of Theorem 3.1. We start showing the theorem for two particular cases. First, if the unbounded face has only two edges on its boundary, then as argued in the proof of Proposition 3.2 we have f0 = 1 as well as v0 = 2 and n ≥ 2, so that the theorem holds. Second, let us consider the case when there is a 0-face F with more than three edges such that only three pseudocircles γi , γj , γk ∈ Γ contribute edges to ∂F . Then the same face F will appear in the subarrangement {γi , γj , γk }. As can be seen from the classification of complete arrangements of three pseudocircles in Figure 2, F must be the unbounded face of a β-arrangement, as the 0-faces of all other arrangements have at most three edges. Consequently, F is the only face of weight 0 in Γ and we have f0 = 1, v0 = 4 as well as n ≥ 3, so that the theorem holds. Thus, let us now assume that the unbounded 0-face and therefore all 0-faces have at least three edges on its boundary and that all 0-faces with more than three edges have more than three pseudocircles contributing to its boundary. Let f0,i be the number of 0faces with i edges. We prove the theorem by induction on the tuples (f0,n , f0,n−1 , . . . , f0,4 ) in lexicographical order. First, if all f0,i with i > 3 are 0, then all 0-faces are triangles, so that v0 = 3f0 and by Proposition 3.2, v0 = 2f0 + f0 ≤ 2f0 + 2n − 4. If there is at least one 0-face F with more than three edges, we may add a pseudocircle γ which intersects for each pseudocircle contributing to ∂F one boundary edge of F (cf. Figure 4, which shows the case when each pseudocircle contributes one edge to ∂F ). By assumption, more than three pseudocircles contribute edges to F , so that in the resulting arrangement Γ′ there is an f0,i (i > 3) being smaller than in Γ, while only f0,j with i > j may have increased. Therefore, the induction assumption is applicable. Now, if f0 has increased by ℓ, then v0 has increased by 2(ℓ + 1), so that by induction assumption v0 (Γ) + 2(ℓ + 1) = v0 (Γ′ ) ≤ 2(n + 1) + 2 f0 (Γ) + ℓ − 4, whence the theorem follows.
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Theorem 1.6 now is an immediate consequence of the following bound on f0 (for a proof we refer to Section 4) together with Theorem 3.1. Theorem 3.3. In complete, α4 -free arrangements of n ≥ 2 pseudocircles, f0 ≤ n − 1. 4. Proof of Theorem 3.3 4.1. Preliminaries. For α-arrangements Γα we denote the bounded 0-face of Γα by Fˆ0 (Γα ). Sloppily, we will often say that a point or face is inside Γα , meaning that it is contained in Fˆ0 (Γα ). Correspondingly, by saying that something is outside Γα , we mean that it is contained in the unbounded 0-face of Γα . We start with some simple facts about the topology of (complete) arrangements. The aim is to establish that in each complete arrangement with more than one 0-face there is an α-arrangement Γα , such that all bounded 0-faces are contained in Fˆ0 (Γα ). Observation 4.1. Let Γ be an arrangement of pseudocircles and γ ∈ Γ. Then two 0faces F1 , F2 of Γ are merged into one when removing γ if and only if γ has an edge ei of weight 0 on ∂Fi (i = 1, 2) and e1 and e2 can be connected by a curve only contained in the interior of γ (and especially not intersecting any other pseudocircle). In particular, such 0-faces F1 , F2 exist when f0 (Γ) > f0 (Γ \ {γ}). In the following, we will say that γ separates F or that γ separates F1 from F2 , if by removing γ from Γ two 0-faces F1 , F2 are merged into a 0-face F in Γ \ {γ} with F1 , F2 ⊆ F . Proposition 4.2. Let Γ be a complete arrangement of pseudocircles with f0 (Γ) > 1. Then Γ has an α-subarrangement. Proof. We show that for α-free arrangements f0 = 1. Assume that there is an α-free arrangement Γ with f0 (Γ) > 1. By Theorem 2.1, fn (Γ) = 1, so that we may apply Proposition 2.2 to conclude that there is an arrangement Γ′ with fn (Γ′ ) > 1. But this contradicts Theorem 2.1. Lemma 4.3 (Separation Lemma). Let F + , F − be two 0-faces in some complete arrangement Γ of pseudocircles. Then there is an α-arrangement Γα ⊆ Γ such that F + is inside Γα , while F − is outside Γα (or vice versa). Proof. First remove pseudocircles from Γ one after another, such that F + and F − are not merged. When this is no longer possible, then in the resulting arrangement Γ′ = {γ1 , . . . , γk } each remaining γi separates F + from F − . That is, on the boundary of each − + + − γi there are two 0-edges e+ and e− i and ei such that ei is on ∂F i is on ∂F . By Proposition 4.2, Γ′ contains some α-arrangement Γα = {γi1 , γi2 , γi3 }. Assume that both − + − + − F + and F − are contained in the same 0-face of Γα . Then the edges e+ i1 , ei1 , ei2 , ei2 , ei3 , ei3 are all contained on the same boundary curve C of Γα . However, it is not possible to − + distribute the edges e+ j , ej on the pseudocircles γj (j = i1 , i2 , i3 ) such that the ej can be
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e+ i
γi
e+ j
e− i Fˆ0 (Γα )
F−
F+
γj e− j
Figure 5. Illustration for the proof of Lemma 4.4 when |Γ∗ | ≥ 2. + separated from the e− and F − cannot be contained in the same j on C. It follows that F 0-face of Γα .
Lemma 4.4. Let F + , F − be two 0-faces in some complete arrangement Γ that are both inside some α-arrangement Γα ⊆ Γ. Then there is another α-arrangement Γ+ α ⊆ Γ with + + − + F inside Γα , and F outside Γα , such that all 0-faces of Γ that are inside Γ+ α are also inside Γα . Proof. By the Separation Lemma 4.3 there is an α-arrangement Γ′α that separates F + from F − . Remove pseudocircles from Γ′α as long as the faces F + , F − remain separated, and let Γ∗ be a minimal subset of Γ′α such that when removing a pseudocircle in Γ∗ from Γ∗ ∪ Γα the faces F + , F − are merged. (Here and in the following F + and F − actually denote the possibly larger 0-faces in Γ∗ ∪ Γα containing the original faces F + and F − .) If Γ∗ contains only a single element γ then F + and F − are separated in Γα ∪ {γ}, and it is + easy to see that γ forms with suitable pseudocircles in Γα an α-arrangement Γ+ α with F ˆ inside and F − outside. Furthermore, Fˆ0 (Γ+ α ) ⊆ F0 (Γα ) so that all 0-faces of Γ that are inside Γ+ α are also inside Γα . Now assume that |Γ∗ | ≥ 2. As no pseudocircle ∈ Γ∗ can be removed without merging − F + and F − , for each pseudocircle γi ∈ Γ∗ there are two 0-edges e+ i , ei on the boundary − − + + − of γi such that e+ i is on ∂F , ei is on ∂F , and ei can be connected with ei by a + − curve only contained in int(γi ). Now, ei and ei must lie on the same component Ci of + γi ∩ Fˆ0 (Γα ): Otherwise either e− i cannot be connected with ei by a curve only contained in int(γi ), or F + and F − are separated in Γα ∪ {γi }, so that one could choose Γ∗ = {γi }. − Furthermore, e+ i and ei obviously cannot be neighbors on Ci , so that there must be a − γj ∈ Γ∗ that separates e+ i from ei on Ci . Thus, by completeness of the arrangement, the situation is as shown in Figure 5. Note that by Observation 4.1, γj ∩ ∂ Fˆ0 (Γα ) cannot lie − in the interior of γi , as in this case it would be impossible to connect the edges e+ i , ei by a curve only contained in the interior of γi . Similarly, γi ∩ ∂ Fˆ0 (Γα ) cannot lie in the interior of γj . Thus, for each γ ∈ Γα it holds that neither int(γ) contains a point from γi ∩ γj , nor can either int(γi ) or int(γj ) contain a point of γ ∩ γj or γ ∩ γi , respectively. It follows that γi , γj together with any γ ∈ Γα forms an α-arrangement Γγ such that all 0-faces of Γ that are inside Γγ are also inside Γα . Moreover, it is easy to see that for a suitable γ the 0-face F + is inside Γγ while F − is outside Γγ . Repeated application of Lemma 4.4 gives the following result.
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Corollary 4.5. Let Γ be a complete arrangement with bounded 0-face F . Then there is an α-subarrangement Γα in Γ such that F is the only 0-face inside Γα . Projecting the arrangement to the sphere the unbounded face of weight 0 becomes bounded, while Corollary 4.5 is still applicable, which therefore yields the following counterpart to Corollary 4.5. Corollary 4.6. Given a complete arrangement Γ with f0 (Γ) > 1, there is an α-subarrangement Γα in Γ such that all bounded 0-faces are inside Γα .
4.2. Proof of Theorem 3.3. We conclude with the following lemma from which Theorem 3.3 follows immediately. Lemma 4.7. Let Γ be a complete, α4 -free arrangement with f0 (Γ) > 1, and let Γα be an α-subarrangement of Γ that contains all bounded 0-faces in its interior (cf. Corollary 4.6). Then there is a γ ∈ Γα such that removing γ from Γ loses at most one face of weight 0, i.e., f0 (Γ \ {γ}) ≥ f0 (Γ) − 1. Proof of Theorem 3.3. The theorem is trivial for n = 2, while for n = 3 it can be verified from Figure 2. We proceed by induction on n. If f0 (Γ) ≤ 1 the theorem again holds trivially for n ≥ 2. If f0 (Γ) > 1 then Lemma 4.7 allows to remove a pseudocircle γ, so that f0 is decreased by at most 1. Together with the induction assumption for Γ \ {γ} this shows the theorem. Proof of Lemma 4.7. We derive a contradiction from the assumption that for each γi ∈ Γα = {γ1 , γ2, γ3 }, f0 (Γ \ {γi }) < f0 (Γ) − 1 (i.e., removing any pseudocircle in Γα from Γ decreases f0 by at least 2). Under this assumption, each γi ∈ Γα borders on at least three 0-faces, two of which (by assumption on Γα ) are contained in Fˆ0 (Γα ). According to Observation 4.1 these two faces Fi,1 , Fi,2 can be connected by a curve Ci only contained in the interior of γi . Each Ci separates the interior of γi into two regions Ri− , Ri+ , where we assume Ri+ to be the region that borders on the unbounded face of weight 0, see Figure 6. On the other hand, Fi,1 , Fi,2 must be separated by some other pseudocircle: For each pseudocircle γi in Γα there is a pseudocircle γi′ ∈ / Γα that intersects γi in Ri− (i = 1, 2, 3). Any such γi′ separates Fi,1 from Fi,2 . Note that both {γi , γi′ , γj } (j 6= i) and {γi , γi′ , γk } (k 6= i, j) are α-arrangements, since γi′ cannot intersect γj (j 6= i) in the interior of γi without crossing Ci (which cannot happen according to Observation 4.1). Summarizing, one of the two 0-faces Fi,1 , Fi,2 is contained inside an α-arrangement {γi , γi′ , γj }, the other contained inside an α-arrangement {γi , γi′ , γk }, cf. Figure 6.
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+ + + + + + + . . . . + . . . . . + . . . .. . . .. . + + . . . + . . . . . . . + + . +
γ1
+
γ1′
+ +
γ3′
+
+
+ + +
+ +
γ2
+.
γ2′
+ + + + + + + . . . . .. . . . . + .. ... . . .. . . + . . .. . + . . . .. + . + + + +
. + ... . . . . .. .. . ... . . .. . . . . . . + + . .. .. . . . . . + + . . . . + + +
γ3
+
Figure 6. Illustration of the situation in Lemma 4.7. The curves Ci separate int(γi ) into regions Ri+ and Ri− (dotted). The pseudocircles γi′ intersect γi in Ri− . + + + + + + + . . . . + . . . . . + . . . .. . . .. . + + . . . + . . . . . . . + + . +
γ1
+
+
γj′
+ +
+
+ + +
+
+.
+ +
γ2
+ + + + + + + . . . . .. . . . . + . .. .. . . .. . . + . . .. . + . . . .. + . + + + +
. + ... . . . . .. .. . ... . . .. . . . . . . + . . . . . . . . + . . . . . . + + + +
γ3
+
Figure 7. If γj′ intersects Ri− , γ3 can be intersected only by allowing an α4 -arrangement. We distinguish the following two cases: ⊲ |{γ1′ , γ2′ , γ3′ }| = 1: In this case γ1′ = γ2′ = γ3′ , so that Γα ∪ {γ1′ } is an α4 -arrangement, contrary to our assumption. ⊲ |{γ1′ , γ2′ , γ3′ }| ≥ 2: Assume without loss of generality that γ1′ 6= γ2′ . Let us consider the intersections of γj′ with γi for i, j ∈ {1, 2} and i 6= j. First note that int(γj′ ) cannot intersect both Ri+ and Ri− , as this would cause a forbidden intersection with Ci . Now, if γj′ intersects Ri− , γj′ cannot intersect γ3 in the interior of either γi or γj without intersecting either Ci or Cj . Therefore, in this case {γi , γj , γ3 , γj′ } is a forbidden α4 -arrangement, cf. Figure 7. Thus let us assume that γ2′ intersects R1+ and γ1′ intersects R2+ , and consequently int(γ2′ ) ∩ R1− = ∅ as well as int(γ1′ ) ∩ R2− = ∅. Since we also have int(γi′ ) ∩ Ri+ = ∅ for i = 1, 2, it follows that int(γ1′ ) ∩ int(γ2′ ) ∩ int(γ1 ) = ∅ and int(γ1′ ) ∩ int(γ2′ ) ∩ int(γ2 ) = ∅. Therefore {γ1′ , γ2′ , γ1} and {γ1′ , γ2′ , γ2 } are α-arrangements. We have already argued above that {γ1 , γ1′ , γ2} and {γ2 , γ2′ , γ1 } are α-arrangements as well, so that {γ1 , γ2, γ1′ , γ2′ } is a forbidden α4 -arrangement.
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5. Discussion and Open Problems 5.1. Consequences. The improved upper bound on v0 of Theorem 1.6 can in turn be used to improve the upper bound on v≤k for complete, α4 -free arrangements. Theorem 5.1. For complete, α4 -free arrangements of n ≥ 2 pseudocircles and k > 0, v≤k ≤ 18kn. Proof. The proof is basically identical to the proof of Theorem 1.3 in [9], only with the application of Theorem 1.2 replaced by an application of Theorem 1.6 and the constants adapted accordingly. Remark 5.2. As α4 -arrangements cannot be realized with unit circles, Theorems 1.6 and 5.1 hold in particular for complete arrangements of unit circles. Note that for general arrangements of unit circles no significant improvement of Theorem 1.2 is possible: Consider the arrangement obtained from a densest circle packing by increasing the radius of the circles by a sufficiently small value, so that each touching point is transformed into two intersection points of weight 0. Then each circle intersects six others, which gives a total of ≈ 6n vertices of weight 0. 5.2. Generalization. A natural question is whether the results of Theorems 1.5 and 1.6 can be generalized to arbitrary (i.e., not necessarily complete) arrangements. We conjecture that a pendant to Theorem 1.5 holds for arrangements that do not contain a cyclic chain according to the following definition. Definition 5.3. A cyclic chain is an arrangement Γ whose elements can be indexed such that Γ = {γ1 , . . . , γn }, and γi intersects γi+1 in two vertices of weight 0 for all i = 1, . . . , n (with γn+1 := γ1 ). Also, there are no further intersections. Our current proof of Theorem 1.5 cannot be easily adapted to show this conjecture. Still, if one is able to generalize the result of Theorem 1.4 to certain arrangements on the sphere (it is easy to see that Theorem 1.4 cannot hold for all arrangements on the sphere), a modification may succeed. Concerning Theorem 1.6, it is not even clear what a generalization may look like. An obvious generalization of an α4 -arrangement would be that of a wheel. Definition 5.4. A wheel is an arrangement which consists of a cyclic chain Γ′ plus another pseudocircle γ which separates one of the two faces of weight 0 in Γ′ into at least three faces of weight 0. Indeed, for arrangements in which all vertices have weight 0, the following result can be considered as a generalization of Theorem 1.6. It follows immediately by application of the following, already mentioned result of C. Thomassen [10] to the arrangement’s intersection graph (as defined after Theorem 1.2): Any graph G = (V, E) that does not contain a semitopological graph S3 (i.e., a cycle plus an additional vertex that is connected to three vertices in the cycle) has at most 2|V | − 3 edges.
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Figure 8. Difficulties when trying to generalize Theorem 1.6. Proposition 5.5. For wheel-free arrangements of n ≥ 2 pseudocircles with no vertices of weight > 0, v0 ≤ 4n − 6. However, Proposition 5.5 does not hold in general, as the arrangement on the left of Figure 8 shows. Although the arrangement does not contain a wheel,1 neither of the bounds f0 ≤ n − 1 and v0 ≤ 4n − 6 holds. On the other hand, simply allowing further intersection points in the definition of cyclic chain is too generous. While it is possible to obtain a variant of Theorem 1.6 for arbitrary arrangements, this result is not a proper generalization of Theorem 1.6: The (complete) arrangement on the right of Figure 8 is α4 -free, but not wheel-free according to the modified definition. Thus, it is not clear what a good generalization of Theorem 1.6 may look like. Acknowledgments. The author would like to thank J. Linhart for helpful comments and suggestions on prior versions of this paper as well as G¨ unter Rote and two anonymous reviewers for pointing out some errors and other suggestions that led to a considerably shorter proof of Lemma 4.7 and general improvement of the paper. A revision of the paper has been prepared while the author was supported by the Austrian Science Fund (FWF): J 3259-N13. References [1] M. Breen, A Helly-type theorem for intersections of compact connected sets in the plane, Geom. Dedicata 71 (1998), 111–117. [2] G. A. Dirac, In abstrakten Graphen vorhandene vollst¨andige 4-Graphen und ihre Unterteilungen, Math. Nachr. 22 (1960), 61–85. [3] K. Kedem, R. Livne, J. Pach, and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1 (1986), 59–71. [4] P. Koebe, Paul , Kontaktprobleme der konformen Abbildung, Ber. Verh. S¨ achs. Akad. Leipzig, Math.-Phys. Kl. 88 (1936), 141–164 . [5] J. Linhart, private communication. [6] J. Linhart, R. Ortner, A note on convex realizability of arrangements of pseudocircles. Geombinatorics XVIII/2 (2008), 66–71. 1As
the two circles intersect (inside the ellipse), the pseudocircles outside do not form a cyclic chain as defined in Definition 5.3.
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[7] J. Linhart, Y. Yang, Arrangements of arcs and pseudocircles, Beitr¨ age Algebra Geom. 37/2 (1996), 391–398. ¨ [8] J. Moln´ ar, Uber den zweidimensionalen topologischen Satz von Helly (in Hungarian, Russian and German summary), Mat. Lapok 8 (1957), 108–114. [9] M. Sharir, On k-sets in arrangements of curves and surfaces, Discrete Comput. Geom. 6 (1991), 593–613. [10] C. Thomassen A minimal condition implying a special K4 -subdivision in a graph, J. Arch. Math. (Basel) 25 (1974), 210–215. [11] Y. Yang, Arrangements of circles on E 2 , unpublished typescript.
E-mail address: [email protected] ¨t Leoben Department Mathematik und Informationstechnologie, Montanuniversita Franz-Josef-Strasse 18, A-8700 Leoben, Austria
