Multitriangulations, Pseudotriangulations and Primitive Sorting Networks
 221 Downloads
 11 Citations
Abstract
We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial.
As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.
Keywords
Pseudoline arrangement Pseudotriangulation Multitriangulation Flip Sorting network Enumeration algorithm1 Introduction
The original motivation for this paper is the interpretation of certain families of planar geometric graphs in terms of pseudoline arrangements. As an introductory illustration, we present this interpretation on the family of triangulations of a convex polygon.
We now switch to the line space \(\mathcal {M}\) of the Euclidean plane ℝ^{2}. Remember that \(\mathcal {M}\) is a Möbius strip homeomorphic to the quotient space ℝ^{2}/(θ,d)∼(θ+π,−d) via the parametrization of an oriented line by its angle θ with the horizontal axis and its algebraic distance d to the origin (see Sect. 3 for details). For pictures, we represent \(\mathcal {M}\) as a vertical band whose boundaries are identified in opposite directions. The dual of a point p∈P is the set of all lines of ℝ^{2} passing through p. It is a pseudoline of \(\mathcal {M}\), i.e. a nonseparating simple closed curve. The dual of P is the set P ^{∗}:={p ^{∗}∣p∈P}. It is a pseudoline arrangement of \(\mathcal {M}\), since any two pseudolines p ^{∗},q ^{∗} of P ^{∗} cross precisely once at the line (pq). Call first level of P ^{∗} the boundary of the external face of the complement of P ^{∗}. It corresponds to the supporting lines of the convex hull of P. See Fig. 1(b).
 (i)
the set Δ^{∗} of all bisectors of a triangle Δ of T is a pseudoline of \(\mathcal {M}\);
 (ii)
the dual pseudolines \(\Delta_{1}^{*},\Delta_{2}^{*}\) of any two triangles Δ_{1},Δ_{2} of T have a unique crossing point (the unique common strict bisector of Δ_{1} and Δ_{2}) and possibly a contact point (when Δ_{1} and Δ_{2} share a common edge);
 (iii)
the set T ^{∗}:={Δ^{∗}∣Δ triangle of T} is a pseudoline arrangement with contact points; and
 (iv)
T ^{∗} covers P ^{∗} minus its first level.
We furthermore prove in this paper that this interpretation is bijective: any pseudoline arrangement with contact points supported by P ^{∗} minus its first level is the dual pseudoline arrangement T ^{∗} of a triangulation T of P.
Motivated by this interpretation, we study the set of all pseudoline arrangements with contact points which cover a given support in the Möbius strip. We define a natural notion of flip between them, and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support (similar to the enumeration algorithm of [7] for pseudotriangulations), based on certain greedy pseudoline arrangements and their connection with primitive sorting networks [23, Sect. 5.3.4, 10, 24]. The running time per arrangement and the working space of our algorithm are both polynomial.
 Pseudotriangulations (k=1).

Introduced for the study of the visibility complex of a set of disjoint convex obstacles in the plane [34, 35], pseudotriangulations were used in different contexts such as motion planning and rigidity theory [15, 42]. Their combinatorial and geometric structure has been extensively studied in the last decade (number of pseudotriangulations [1, 2], polytope of pseudotriangulations [36], algorithmic issues [5, 7, 16], etc.). See [37] for a detailed survey on the subject. As far as pseudotriangulations are concerned, this paper has two main applications: it proves the dual characterization of pseudotriangulations in terms of pseudoline arrangements and provides an interpretation of greedy pseudotriangulations in terms of sorting networks, leading to a new proof of the greedy flip property for points [3, 7, 34]. The objects studied in this paper have a further (algorithmic) motivation: as a first step to compute the dual arrangement of a set of disjoint convex bodies described only by its chirotope, Habert and Pocchiola raise in [16] the question to compute efficiently a pseudotriangulation of a pseudotriangulation of the set, i.e. a 2pseudotriangulation.
 Multitriangulations (convex position).

Introduced in the context of extremal theory for geometric graphs [9], multitriangulations were then studied for their combinatorial structure [11, 12, 20, 25]. The study of stars in multitriangulations [28], generalizing triangles for triangulations, naturally leads to interpret multitriangulations as multipseudotriangulations of points in convex position. As far as we know, this paper provides the first interpretation of multitriangulations in terms of pseudoline arrangements on the Möbius strip.
The paper is organized as follows. Section 2 is devoted to the study of all pseudoline arrangements with contact points covering a given support. We define the flip and the greedy pseudoline arrangements (2.2) whose properties yield the enumeration algorithm for pseudoline arrangements with a given support (2.3).
In Sect. 3, we prove that the pseudotriangulations of a finite planar point set P in general position correspond to the pseudoline arrangements with contact points supported by the dual pseudoline arrangement of P minus its first level (3.2). Similarly, we observe that multitriangulations of a planar point set P in convex position correspond to pseudoline arrangements with contact points supported by the dual pseudoline arrangement of P minus its first k levels (3.3).
This naturally yields to the definition of multipseudotriangulations in Sect. 4. We study the primal of a multipseudotriangulation. We discuss some of its structural properties (4.2) which generalize the cases of pseudotriangulations and multitriangulations: number of edges, pointedness, crossingfreeness. We study in particular the faces of multipseudotriangulations (4.3) which naturally extend triangles in triangulations, pseudotriangles in pseudotriangulations, and stars in multitriangulations.
In Sect. 5, we compare multipseudotriangulations to iterated pseudotriangulations. We give an example of a 2triangulation which is not a pseudotriangulation of a triangulation (5.1). We prove, however, that greedy multipseudotriangulations are iterated greedy pseudotriangulations (5.2), and we study flips in iterated pseudotriangulations (5.3).
Section 6 presents two further topics. The first one (6.1) is a pattern avoiding characterization of greedy multipseudotriangulations related to horizon trees. The second one (6.2) is a discussion on multipseudotriangulations of double pseudoline arrangements, which extend pseudotriangulations of convex bodies in the plane.
Finally, we discuss in Sect. 7 some related problems and open questions concerning in particular the primal of a multipseudotriangulation, the diameter and the polytopality of the graph of flips, and the number of multipseudotriangulations. Since the submission of this paper, some of these questions were partially answered in [29, 40, 43] based on a framework similar to the material presented in this paper.
2 Pseudoline Arrangements with the Same Support
2.1 Pseudoline Arrangements in the Möbius Strip
Let \(\mathcal {M}\) denote the Möbius strip (without boundary), defined as the quotient set of the plane ℝ^{2} under the map \(\tau: \mathbb {R}^{2}\to \mathbb {R}^{2},\; (x,y)\mapsto(x+\pi,y)\). The induced canonical projection will be denoted by \(\pi:\mathbb {R}^{2}\to \mathcal {M}\).
A pseudoline is the image λ under the canonical projection π of the graph {(x,f(x))∣x∈ℝ} of a continuous and πantiperiodic function f:ℝ→ℝ (that is, which satisfies f(x+π)=−f(x) for all x∈ℝ). We say that f represents the pseudoline λ.
When we consider two pseudolines, we always assume that they have a finite number of intersection points. Thus, these intersection points can only be either crossing points or contact points. See Fig. 3(a). Any two pseudolines always have an odd number of crossing points (in particular, at least one). When λ and μ have exactly one crossing point, we denote it by λ∧μ.
Remark 1
The usual definition of pseudoline arrangements does not allow contact points. In this paper, they play a crucial role since we are interested in all pseudoline arrangements which share a common support, and which only differ by their sets of contact points. To simplify the exposition, we omit throughout the paper to specify that we work with pseudoline arrangements with contact points.
Pseudoline arrangements are also classically defined on the projective plane rather than the Möbius strip. The projective plane is obtained from the Möbius strip by adding a point at infinity.
For more details on pseudoline arrangements, we refer to the broad literature on the topic [6, 13, 14, 24].
2.2 Flip Graph and Greedy Pseudoline Arrangements
2.2.1 Flips
Lemma 2
Let Λ be a pseudoline arrangement, \(\mathcal {S}\) be its support, and V be the set of its contact points. Let v∈V be a contact point of two pseudolines of Λ, and w denote their unique crossing point. Then V△{v,w} is also the set of contact points of a pseudoline arrangement Λ′ supported by \(\mathcal {S}\).
Proof
Definition 3
Let Λ be a pseudoline arrangement with support \(\mathcal {S}\) and contact points V, let v∈V be a contact point between two pseudolines of Λ which cross at w, and let Λ′ be the pseudoline arrangement with support \(\mathcal {S}\) and contact points V△{v,w}. We say that we obtain Λ′ by flipping v in Λ.
Note that the starting point of a flip is always a contact point. To this contact point corresponds precisely one crossing point. In contrast, it would be incorrect to try to flip a crossing point: the two pseudolines which cross at this point might have zero or many contact points.
Observe also that the pseudoline arrangements Λ and Λ′ are the only two pseudoline arrangements supported by \(\mathcal {S}\) whose sets of contact points contain V∖{v}.
Definition 4
Let \(\mathcal {S}\) be the support of a pseudoline arrangement. The flip graph of \(\mathcal {S}\), denoted by \(G(\mathcal {S})\), is the graph whose vertices are all the pseudoline arrangements supported by \(\mathcal {S}\), and whose edges are flips between them.
In other words, there is an edge in the graph \(G(\mathcal {S})\) between two pseudoline arrangements if and only if their sets of contact points differ by exactly two points.
Observe that the graph \(G(\mathcal {S})\) is regular: there is one edge adjacent to a pseudoline arrangement Λ supported by \(\mathcal {S}\) for each contact point of Λ, and two pseudoline arrangements with the same support have the same number of contact points.
Example 5
The flip graph of the support of an arrangement of two pseudolines with p contact points is the complete graph on p+1 vertices.
2.2.2 Acyclic Orientations
Let \(\mathcal {S}\) be the support of a pseudoline arrangement and \(\bar{\mathcal {S}}\) denote its preimage under the projection π. We orient the graph \(\bar{\mathcal {S}}\) along the abscissa increasing direction, and the graph \(\mathcal {S}\) by projecting the orientations of the edges of \(\bar{\mathcal {S}}\). We denote by ≼ the induced partial order on the vertex set of \(\bar {\mathcal {S}}\) (defined by z≼z′ if there exists an oriented path on \(\bar {\mathcal {S}}\) from z to z′).
A filter of \(\bar{\mathcal {S}}\) is a proper set F of vertices of \(\bar {\mathcal {S}}\) such that z∈F and z≼z′ implies z′∈F. The corresponding antichain is the set of all edges and faces of \(\bar{\mathcal {S}}\) with one vertex in F and one vertex not in F. This antichain has a linear structure, and thus, can be seen as the set of edges and faces that cross a vertical curve \(\bar{\chi}\) of ℝ^{2}. The projection \(\chi :=\pi(\bar{\chi})\) of such a curve is called a cut of \(\mathcal {S}\). We see the fundamental domain located between the two curves \(\bar{\chi}\) and \(\tau(\bar{\chi})\) as the result of cutting the Möbius strip along the cut χ. For example, we use such a cut to represent pseudoline arrangements in all figures of this paper. See for example Fig. 3.
The cut χ defines a partial order ≼_{ χ } on the vertex set of \(\mathcal {S}\): for all vertices v and w of \(\mathcal {S}\), we write v≼_{ χ } w if there is an oriented path in \(\mathcal {S}\) which does not cross χ. In other words, v≼_{ χ } w if \(\bar{v}\preccurlyeq \bar{w}\), where \(\bar{v}\) (resp. \(\bar{w}\)) denotes the unique preimage of v (resp. w) between \(\bar{\chi}\) and \(\tau(\bar{\chi})\). For example, in the arrangements of Fig. 3, we have v≺_{ χ } w.
Let Λ be a pseudoline arrangement supported by \(\mathcal {S}\), v be a contact point between two pseudolines of Λ and w denote their crossing point. Since v and w lie on a same pseudoline on \(\mathcal {S}\), they are comparable for ≼_{ χ }. We say that the flip of v is χincreasing if v≺_{ χ } w and χdecreasing otherwise. For example, the flip from (a) to (b) in Fig. 3 is χincreasing. We denote by \(G_{\chi}(\mathcal {S})\) the directed graph of χincreasing flips on pseudoline arrangements supported by \(\mathcal {S}\).
Lemma 6
The directed graph \(G_{\chi}(\mathcal {S})\) of χincreasing flips is acyclic.
Proof
If Λ and Λ′ are two pseudoline arrangements supported by \(\mathcal {S}\), we write Open image in new window if there exists a bijection ϕ between their sets of contact points such that v≼_{ χ } ϕ(v) for any contact point v of Λ. It is easy to see that this relation is a partial order on the vertices of \(G_{\chi}(\mathcal {S})\). Since the edges of \(G_{\chi}(\mathcal {S})\) are oriented according to Open image in new window , the graph \(G_{\chi}(\mathcal {S})\) is acyclic. □
Theorem 7 below states that this acyclic graph \(G_{\chi}(\mathcal {S})\) has in fact a unique source, and thus is connected.
2.2.3 Sorting Networks
 (i)
if v _{ i } is a contact point of Λ, then \(\sigma_{i}^{\varLambda }=\sigma_{i+1}^{\varLambda }\);
 (ii)
otherwise, \(\sigma_{i+1}^{\varLambda }\) is obtained from \(\sigma _{i}^{\varLambda }\) by inverting its i ^{□}th and (i ^{□}+1)th entries.
Theorem 7
The directed graph \(G_{\chi}(\mathcal {S})\) has a unique source Γ, characterized by the property that the permutation \(\sigma_{i+1}^{\varGamma }\) is obtained from \(\sigma_{i}^{\varGamma }\) by sorting its i ^{□} th and (i ^{□}+1)th entries, for all i.
Proof
If Γ satisfies the above property, then it is obviously a source of the directed graph \(G_{\chi}(\mathcal {S})\): any flip of Γ is χincreasing since two of its pseudolines cannot touch before they cross.
 (i)
If a<b, then the two pseudolines λ _{ a } and λ _{ b } of Γ already cross before v _{ i }. Consequently, v _{ i } is necessarily a contact point of Γ, which implies that \(\sigma _{i+1}^{\varGamma }(i^{\square})=a\) and \(\sigma_{i+1}^{\varGamma }(i^{\square}+1)=b\).
 (ii)
If a>b, then the two pseudolines λ _{ a } and λ _{ b } of Γ do not cross before v _{ i }. Since Γ is a source of \(G_{\chi}(\mathcal {S})\), v _{ i } is necessarily a crossing point of Γ. Thus \(\sigma_{i+1}^{\varGamma }(i^{\square})=b\) and \(\sigma_{i+1}^{\varGamma }(i^{\square}+1)=a\).
Corollary 8
The graphs of flips \(G(\mathcal {S})\) and \(G_{\chi}(\mathcal {S})\) are connected.
Definition 9
The unique source of the directed graph \(G_{\chi}(\mathcal {S})\) is denoted by \(\varGamma _{\chi}(\mathcal {S})\) and called the χgreedy pseudoline arrangement on \(\mathcal {S}\).
Let us reformulate Theorem 7 in terms of sorting networks (see [23, Sect. 5.3.4] for a detailed presentation; see also [10]). Let i<j be two integers. A comparator [i:j] transforms a sequence of numbers (a _{1},…,a _{ n }) by sorting (a _{ i },a _{ j }), i.e. replacing a _{ i } by min(a _{ i },a _{ j }) and a _{ j } by max(a _{ i },a _{ j }). A comparator [i:j] is primitive if j=i+1. A sorting network is a sequence of comparators that sorts any sequence (a _{1},…,a _{ n }).
The support \(\mathcal {S}\) of an arrangement of n pseudolines together with an arbitrary sweep F _{ m }⊃⋯⊃F _{0} corresponds to the primitive sorting network [1^{□}:1^{□}+1],…,[m ^{□}:m ^{□}+1] (see [24, Sect. 8]). Theorem 7 affirms that sorting the permutation [n,n−1,…,2,1] according to this sorting network provides a pseudoline arrangement supported by \(\mathcal {S}\), which depends only upon the support \(\mathcal {S}\) and the filter F _{0} (not on the total order given by the sweep).
2.2.4 Greedy Set of Contact Points
The following proposition provides an alternative construction of the greedy pseudoline arrangement \(\varGamma _{\chi}(\mathcal {S})\) of the support \(\mathcal {S}\).
Proposition 10
Let v _{1},…,v _{ q } be a sequence of vertices of \(\mathcal {S}\) constructed recursively by choosing, as long as possible, a remaining vertex v _{ i } of \(\mathcal {S}\) minimal (for the partial order ≼_{ χ }) such that {v _{1},…,v _{ i }} is a subset of the set of contact points of a pseudoline arrangement supported by \(\mathcal {S}\). Then the resulting set {v _{1},…,v _{ q }} is exactly the set of contact points of \(\varGamma _{\chi}(\mathcal {S})\).
Proof
First of all, {v _{1},…,v _{ q }} is by construction the set of contact points of a pseudoline arrangement Λ supported by \(\mathcal {S}\). If Λ is not the (unique) source \(\varGamma _{\chi}(\mathcal {S})\) of the oriented graph \(G_{\chi}(\mathcal {S})\), then there is a contact point v _{ i } of Λ whose flip is χdecreasing. Let w denote the corresponding crossing point, and Λ′ the pseudoline arrangement obtained from Λ by flipping v _{ i }. This implies that {v _{1},…,v _{ i−1},w} is a subset of the set of contact points of Λ′ and that w≺_{ χ } v _{ i }, which contradicts the minimality of v _{ i }. □
 (i)
sweeping \(\mathcal {S}\) decreasingly and place crossing points as long as possible; or
 (ii)
sweeping \(\mathcal {S}\) increasingly and place contact points as long as possible.
2.2.5 Constrained Flip Graph
We now need to extend the previous results to constrained pseudoline arrangements on \(\mathcal {S}\), in which we force a set V of vertices of \(\mathcal {S}\) to be contact points.
Theorem 11
 (i)
if v _{ i }∈V, then \(\sigma_{i+1}^{\varGamma }=\sigma_{i}^{\varGamma }\);
 (ii)
if v _{ i }∉V, then \(\sigma_{i+1}^{\varGamma }\) is obtained from \(\sigma_{i}^{\varGamma }\) by sorting its i ^{□} th and (i ^{□}+1)th entries.
Proof
We transform our support \(\mathcal {S}\) into another one \(\mathcal {S}'\) by opening all intersection points of V (the local picture of this transformation is Open image in new window ). If \(\mathcal {S}'\) supports at least one pseudoline arrangement, we apply the result of Theorem 7: a pseudoline arrangement supported by \(\mathcal {S}'\) corresponds to a pseudoline arrangement with support \(\mathcal {S}\) whose set of contact points contains V. □
We denote by \(\varGamma _{\chi}(\mathcal {S}\,\,V)\) the unique source of the constrained flip graph \(G_{\chi}(\mathcal {S}\,\,V)\).
In terms of sorting networks, \(\varGamma _{\chi}(\mathcal {S}\,\,V)\) is the result of the sorting of the inverted permutation [n,n−1,…,2,1] by the restricted primitive network ([i ^{□}:i ^{□}+1])_{ i∈I }, where I:={i∣v _{ i }∉V}.
Observe also that we can obtain, like in the previous subsection, the contact points of \(\varGamma _{\chi}(\mathcal {S}\,\, V)\) by an iterative procedure: we start from the set V and add recursively a minimal (for the partial order ≼_{ χ }) remaining vertex v _{ i } of \(\mathcal {S}\) such that V∪{v _{1},…,v _{ i }} is a subset of the set of contact points of a pseudoline arrangement supported by \(\mathcal {S}\). The vertex set produced by this procedure is the set of contact points of the χgreedy constrained pseudoline arrangement \(\varGamma _{\chi}(\mathcal {S}\,\, V)\).
2.3 Greedy Flip Property and Enumeration
2.3.1 Greedy Flip Property
Theorem 12
(Greedy flip property)
 (1)
If v is a contact point of \(\varGamma _{\chi}(\mathcal {S}\,\,V)\) which is not in V, then \(\varGamma _{\psi}(\mathcal {S}\,\,V)\) is obtained from \(\varGamma _{\chi}(\mathcal {S}\,\,V)\) by flipping v. Otherwise, \(\varGamma _{\psi}(\mathcal {S}\,\,V)=\varGamma _{\chi}(\mathcal {S}\,\,V)\).
 (2)
If v is a contact point of \(\varGamma _{\chi}(\mathcal {S}\,\,V)\), then \(\varGamma _{\psi}(\mathcal {S}\,\,W)=\varGamma _{\chi}(\mathcal {S}\,\,V)\). Otherwise, \(G(\mathcal {S}\,\,W)\) is empty.
Proof
 (i)
σ _{1} is the inverted permutation [n,n−1,…,2,1];
 (ii)
if v _{ i }∈V, then σ _{ i+1}=σ _{ i };
 (iii)
otherwise, σ _{ i+1} is obtained from σ _{ i } by sorting its i ^{□}th and (i ^{□}+1)th entries.
 (i)
for all 1≤i≤j, ρ _{ i } is obtained from σ _{ i } by exchanging m ^{□} and m ^{□}+1;
 (ii)
for all j<i≤m, ρ _{ i }=σ _{ i }.

When v is not a crossing point of \(\varGamma _{\chi}(\mathcal {S}\,\, V)\), or is in V, ρ _{ i }=σ _{ i } for all i∈[m], and \(\varGamma _{\psi}(\mathcal {S}\,\, V)=\varGamma _{\chi}(\mathcal {S}\,\, V)\).

When v is a contact point of \(\varGamma _{\chi}(\mathcal {S}\,\, V)\), ω _{ i }=σ _{ i } for all i∈[m], and \(\varGamma _{\psi}(\mathcal {S}\,\, W)= \varGamma _{\chi}(\mathcal {S}\,\, V)\).
Finally, we prove that \(G(\mathcal {S}\,\, W)\) is empty when v is not a contact point of \(\varGamma _{\chi}(\mathcal {S}\,\, V)\). For this, assume that \(G(\mathcal {S}\,\, W)\) is not empty, and consider the greedy arrangement \({\varGamma =\varGamma _{\chi}(\mathcal {S}\,\, W)}\). The flip of any contact point of Γ not in W is χincreasing. Furthermore, since v is a minimal element for ≼_{ χ }, the flip of v is also χincreasing. Consequently, Γ is a source in the graph \(G_{\chi}(\mathcal {S}\,\, V)\), which implies that \(\varGamma _{\chi}(\mathcal {S}\,\, V)=\varGamma \), and thus, v is a contact point of \(\varGamma _{\chi}(\mathcal {S}\,\, V)\). □
2.3.2 Enumeration
From the greedy flip property, we derive a binary tree structure on colored pseudoline arrangements supported by \(\mathcal {S}\), whose leftpending leaves are precisely the pseudoline arrangements supported by \(\mathcal {S}\). A pseudoline arrangement is colored if its contact points are colored in blue, green or red. Green and red contact points are considered to be fixed, while blue ones can be flipped.
Theorem 13
 (i)
The root of the tree is the χgreedy pseudoline arrangement on \(\mathcal {S}\), entirely colored in blue.
 (ii)
Any node Λ of \(\mathcal {T}\) is a leaf of \(\mathcal {T}\) if either it contains a green contact point or it only contains red contact points.
 (iii)
Otherwise, choose a minimal blue point v of Λ. The right child of Λ is obtained by flipping v and coloring it in blue if the flip is χincreasing and in green if the flip is χdecreasing. The left child of Λ is obtained by changing the color of v into red.
Proof
The proof is similar to that of Theorem 9 in [7].
We define inductively a cut χ(Λ) for each node Λ of \(\mathcal {T}\): the cut of the root is χ, and for each node Λ the cut of its children is obtained from χ by sweeping the contact point v. We also denote V(Λ) the set of red contact points of Λ. With these notations, the greedy flip property (Theorem 12) ensures that \(\varLambda =\varGamma _{\chi(\varLambda )}(\mathcal {S}\,\,V(\varLambda ))\), for each node Λ of \(\mathcal {T}\).

the set V(Λ _{ i }) is a subset of contact points of Λ;

the contact points of Λ not in V(Λ _{ i }) are not located between χ(Λ) and χ;
Visiting the tree \(\mathcal {T}\) provides an algorithm to enumerate all pseudoline arrangements with a given support. In the next section, we will see the connection between this algorithm and the enumeration algorithm of [7] for pseudotriangulations of a point set.
Let us briefly discuss the complexity of this algorithm. We assume that the input of the algorithm is a pseudoline arrangement and we consider a flip as an elementary operation. Then this algorithm requires a polynomial running time per pseudoline arrangement supported by \(\mathcal {S}\). As for many enumeration algorithms, the crucial point of this algorithm is that its working space is also polynomial (while the number of pseudoline arrangements supported by \(\mathcal {S}\) is exponential).
3 Dual Pseudoline Arrangements
In this section, we prove that both the graph of flips on “(pointed) pseudotriangulations of a point set” and the graph of flips on “multitriangulations of a convex polygon” can be interpreted as graphs of flips on “pseudoline arrangements with a given support”. This interpretation is based on the classical duality that we briefly recall in the first subsection, and leads to a natural definition of “multipseudotriangulations of a pseudoline arrangement” that we present in Sect. 4.
3.1 Dual Pseudoline Arrangement of a Point Set
To a given oriented line in the Euclidean plane, we associate its angle θ∈ℝ/2πℤ with the horizontal axis and its algebraic distance d∈ℝ to the origin (i.e. the value of 〈(−v,u)∣.〉 on the line, where (u,v) is its unitary direction vector). Since the same line oriented in the other direction gives an angle θ+π and a distance −d, this parametrization naturally associates a point of the Möbius strip \({\mathcal {M}:=\mathbb {R}^{2}/(\theta,d) \sim(\theta+\pi ,d)}\) to each line of the Euclidean plane. In other words, the line space of the Euclidean plane is (isomorphic to) the Möbius strip.
This elementary duality also holds for any topological plane (or ℝ^{2}plane, see [39] for a definition), not only for the Euclidean plane ℝ^{2}. That is to say, the line space of a topological plane is (isomorphic to) the Möbius strip and the dual of a finite set of points in a topological plane is a pseudoline arrangement without contact points. Let us also recall that any pseudoline arrangement of the Möbius strip without contact points is the dual arrangement of a finite set of points in a certain topological plane [17]. Thus, in the rest of this paper, we deal with sets of points and their duals without restriction to the Euclidean plane.
3.2 Dual Pseudoline Arrangement of a Pseudotriangulation
We refer to [37] for a detailed survey on pseudotriangulations, and just recall here some basic definitions.
Definition 14
 (i)
either it passes through a corner of Δ and separates the two edges incident to it;
 (ii)
or it passes through a concave vertex of Δ and does not separate the two edges incident to it.
A pseudotriangulation of a point set P in general position is a set of edges of P which decomposes the convex hull of P into pseudotriangles. We moreover always assume that pseudotriangulations are pointed, meaning that there exists a line passing through any point p∈P and defining a halfplane containing all the edges incident to p.
The results of this paper only concern pointed pseudotriangulations. Therefore we omit to always specify that pseudotriangulations are pointed. Historically, pseudotriangulations were introduced for families of smooth convex bodies [35] and were therefore automatically pointed. Pseudotriangulations of points, pointed or not, can be regarded as limits of pseudotriangulations of infinitesimally small convex bodies. Note that pointed pseudotriangulations are edgeminimal pseudotriangulations.
Observation 15
 (i)
the set Δ^{∗} of all tangents to a pseudotriangle Δ of T is a pseudoline;
 (ii)
the dual pseudolines \(\Delta_{1}^{*}, \Delta_{2}^{*}\) of any two pseudotriangles Δ_{1},Δ_{2} of T have a unique crossing point (the unique common tangent to Δ_{1} and Δ_{2}) and possibly a contact point (when Δ_{1} and Δ_{2} share a common edge);
 (iii)
the set T ^{∗}:={Δ^{∗}∣Δ pseudotriangle of T} is a pseudoline arrangement (with contact points); and
 (iv)
T ^{∗} is supported by P ^{∗} minus its first level (see Fig. 9(b)).
In fact, this covering property characterizes pseudotriangulations:
Theorem 16
 (i)
The dual arrangement T ^{∗}:={Δ^{∗}∣Δ pseudotriangle of T} of a pseudotriangulation T of P is supported by P ^{∗1}.
 (ii)The primal set of edgesof a pseudoline arrangement Λ supported by P ^{∗1} is a pseudotriangulation of P.$$E :=\bigl \{[p,q] \,\big \,p,q\in P,\; p^*\wedge q^* \text{ contact\ point\ of\ } \varLambda \bigr \}$$
Proof 1 of Theorem 16(ii)
The two notions of flips (the primal notion on pseudotriangulations of P and the dual notion on pseudoline arrangements supported by P ^{∗1}) coincide via duality: an internal edge e of a pseudotriangulation T of P corresponds to a contact point e ^{∗} of the dual pseudoline arrangement T ^{∗}; the two pseudotriangles Δ_{1} and Δ_{2} of T containing e correspond to the two pseudolines \(\Delta_{1}^{*}\) and \(\Delta_{2}^{*}\) of T ^{∗} in contact at e ^{∗}; and the common tangent f of Δ_{1} and Δ_{2} corresponds to the crossing point f ^{∗} of \(\Delta_{1}^{*}\) and \(\Delta_{2}^{*}\).
Thus, the graph G(P) is a subgraph of \(G(\mathcal {S})\). Since both are connected and regular of degree P−3, they coincide. In particular, any pseudoline arrangement supported by P ^{∗1} is the dual of a pseudotriangulation of P. □
Remark 17
Observe that this duality matches our greedy pseudoline arrangement supported by P ^{∗1} with the greedy pseudotriangulation of [7]. In particular, the greedy flip property and the enumeration algorithm of Sect. 2.3 are generalizations of results in [7].
Our second proof of Theorem 16 is slightly longer but more direct, and it introduces a “witness method” that we will repeatedly use throughout this paper. It is based on the following characterization of pseudotriangulations:
Lemma 18
[42]
A graph T on P is a pointed pseudotriangulation of P if and only if it is crossingfree, pointed and has 2P−3 edges.
Proof 2 of Theorem 16(ii)
We check that E is crossingfree, pointed and has 2P−3 edges:
 (i)
Since P is in general position, the point t is not in P. Therefore P ^{∗}∪{t ^{∗}}=(P∪{t})^{∗} is a (nonsimple) pseudoline arrangement, and t ^{∗} crosses P ^{∗} exactly P times.
 (ii)
Since t ^{∗} is a pseudoline, it crosses each pseudoline of Λ at least once. Thus, it crosses Λ at least Λ=P−2 times.
 (iii)
For each of the points p ^{∗}∧r ^{∗} and q ^{∗}∧s ^{∗}, replacing the crossing point by a contact point removes two crossings with t ^{∗}.
 (i)
ℓ _{ ε } crosses P ^{∗} exactly P+2 times (it crosses s ^{∗} three times and any other pseudoline of P ^{∗} exactly once).
 (ii)
Since ℓ _{ ε } is a pseudoline, it crosses Λ at least Λ=P−2 times.
 (iii)
For each of the points p ^{∗}∧s ^{∗}, q ^{∗}∧s ^{∗} and r ^{∗}∧s ^{∗}, replacing the crossing point by a contact point removes two crossings with ℓ _{ ε }.
Observe that once we know that E is crossingfree, we could also argue its pointedness observing that the pseudolines of Λ would cover the dual pseudoline of a nonpointed vertex twice.
Proof 3 of Theorem 16(ii)
As a consequence, all polygons S(λ) for λ∈Λ are pseudotriangles (otherwise, we would have points such that σ _{ λ }(q)>1 for some λ) and they cover the convex hull of P. Consequently, these P−2 pseudotriangles form a pointed pseudotriangulation of P. □
3.3 Dual Pseudoline Arrangement of a Multitriangulation
Definition 19
For ℓ∈ℕ, an ℓcrossing is a set of ℓ mutually crossing edges of C _{ n }. A ktriangulation of the ngon is a maximal set of edges of C _{ n } with no (k+1)crossing.
Observe that an edge of C _{ n } can be involved in a (k+1)crossing only if there remain at least k vertices on each side. Such an edge is called krelevant. An edge with exactly (resp. strictly less than) k−1 vertices on one side is a kboundary edge (resp. a kirrelevant edge). By maximality, every ktriangulation consists of all the nk kirrelevant plus kboundary edges and some krelevant edges.
Definition 20
[28]
A kstar is a star polygon of type {2k+1/k}, that is, a set of edges of the form {s _{ j } s _{ j+k }∣j∈ℤ_{2k+1}}, where s _{0},s _{1},…,s _{2k } are cyclically ordered around the unit circle. A (strict) bisector of a kstar is a (strict) bisector of one of its angles s _{ j−k } s _{ j } s _{ j+k }.
Theorem 21
[28]
 (i)
T contains exactly n−2k kstars and k(n−2k−1) krelevant edges.
 (ii)
Each edge of T belongs to zero, one, or two kstars, depending on whether it is kirrelevant, kboundary, or krelevant.
 (iii)
Every pair of kstars of T has a unique common strict bisector.
 (iv)
Flipping any krelevant edge e of T into the common strict bisector f of the two kstars containing e produces a new ktriangulation T△{e,f} of the ngon. T and T△{e,f} are the only two ktriangulations of the ngon containing T∖{e}.
 (v)
The flip graph G _{ n,k } on ktriangulations of the ngon is connected and regular of degree k(n−2k−1).
Observation 22
 (i)
the set S ^{∗} of all bisectors of a kstar S of T is a pseudoline of the Möbius strip;
 (ii)
the dual pseudolines \(S_{1}^{*}, S_{2}^{*}\) of any two kstars S _{1},Δ_{2} of T have a unique crossing point (the unique common strict bisector of S _{1} and S _{2}) and possibly some contact points (when S _{1} and S _{2} share common edges);
 (iii)
the set \(T^{*} :=\{S^{*} \mid S\;k\text{star of } T\}\) of dual pseudolines of kstars of T is a pseudoline arrangement (with contact points); and
 (iv)
T ^{∗} is supported by the dual pseudoline arrangement \(C_{n}^{*}\) of C _{ n } minus its first k levels (see Fig. 15(b)).
Again, it turns out that this observation provides a characterization of multitriangulations of a convex polygon:
Theorem 23
 (i)
The dual pseudoline arrangement \(T^{*} :=\{S^{*} \mid S\;k\text {star of }T\}\) of a ktriangulation T of the ngon is supported by \(C_{n}^{*k}\).
 (ii)The primal set of edgesof a pseudoline arrangement Λ supported by \(C_{n}^{*k}\) is a ktriangulation of the ngon.$$E :=\bigl \{[p,q] \,\big \,p,q\in C_n,\; p^*\wedge q^*\text{ contact point of } \varLambda \bigr \}$$
We provide two proofs of this theorem.
Proof 1 of Theorem 23(ii)
The two notions of flips (the primal notion on ktriangulations of the ngon and the dual notion on pseudoline arrangements supported by \(C_{n}^{*k}\)) coincide. Thus, the flip graph G _{ n,k } on ktriangulations of the ngon is a subgraph of \(G(C_{n}^{*k})\). Since they are both connected and regular of degree k(n−2k−1), these two graphs coincide. In particular, any pseudoline arrangement supported by \(C_{n}^{*k}\) is the dual of a ktriangulation of the ngon. □
Proof 2 of Theorem 23(ii)
We follow the method of our second proof of Theorem 16(ii). Since E has the right number of edges (namely k(2n−2k−1)), we only have to prove that it is (k+1)crossingfree. We consider 2k+2 points p _{0},…,p _{ k },q _{0},…,q _{ k } cyclically ordered around the unit circle. Since the definition of crossing (and thus, of ℓcrossing) is purely combinatorial, i.e. depends only on the cyclic order of the points and not on their exact positions, we can move all the vertices of our ngon on the unit circle while preserving their cyclic order. In particular, we can assume that the lines (p _{ i } q _{ i })_{ i∈{0,…,k}} all contain a common point t. Its dual pseudoline t ^{∗} crosses \(C_{n}^{*}\) exactly n times and Λ at least Λ=n−2k times. Furthermore, for any point \({p_{i}^{*}\wedge q_{i}^{*}}\), replacing the crossing point by a contact point removes two crossings with t ^{∗}. Thus, the pseudoline t ^{∗} provides a witness which proves that the edges [p _{ i },q _{ i }], i∈{0,…,k}, cannot be all in E, and thus ensures that E is (k+1)crossingfree. □
4 Multipseudotriangulations
Motivated by Theorems 16 and 23, we define in terms of pseudoline arrangements a natural generalization of both pseudotriangulations and multitriangulations. We then study elementary properties of the corresponding set of edges in the primal space.
4.1 Definition
We consider the following generalizations of both pseudotriangulations and multitriangulations:
Definition 24
Let L be a pseudoline arrangement supported by \(\mathcal {S}\). Define its kkernel \(\mathcal {S}^{k}\) to be its support minus its first k levels (which are the iterated external hulls of \(\mathcal {S}\)). Denote by V ^{ k } the set of contact points of L in \(\mathcal {S}^{k}\). A kpseudotriangulation of L is a pseudoline arrangement whose support is \(\mathcal {S}^{k}\) and whose set of contact points contains V ^{ k }.
Pseudotriangulations of a point set P correspond via duality to 1pseudotriangulations of the dual pseudoline arrangement P ^{∗}. Similarly, ktriangulations of the ngon correspond to kpseudotriangulations of the pseudoline arrangement \(C_{n}^{*}\) in convex position. If L is a pseudoline arrangement with no contact point, then any pseudoline arrangement supported by \(\mathcal {S}^{k}\) is a kpseudotriangulation of L. In general, the condition that the contact points of L in its kkernel should be contact points of any kpseudotriangulation of L is a natural assumption for iterating multipseudotriangulations (see Sect. 5).
Let Λ be a kpseudotriangulation of L. We denote by V(Λ) the union of the set of contact points of Λ with the set of intersection points of the first k levels of L. In other words, V(Λ) is the set of intersection points of L which are not crossing points of Λ. As for pseudoline arrangements, the set V(Λ) completely determines Λ.
Flips for multipseudotriangulations are defined as in Lemma 2, with the restriction that the contact points in V ^{ k } cannot be flipped. In other words, the flip graph on kpseudotriangulations of L is exactly the graph \(G(\mathcal {S}^{k}\,\, V^{k})\). Section 2 asserts that the graph of flips is regular and connected, and provides an enumeration algorithm for multipseudotriangulations of L.
Definition 25
The χgreedy kpseudotriangulation of L, denoted \(\varGamma _{\chi}^{k}(L)\), is the greedy pseudoline arrangement \(\varGamma _{\chi}(\mathcal {S}^{k}\,\, V^{k})\).
4.2 Pointedness and Crossings
Lemma 26
The primal E of Λ has k(2P−2k−1) edges.
Proof
Lemma 27
The primal E of Λ cannot contain a kalternation.
Proof
 (i)
ℓ crosses P ^{∗} exactly P+2k times;
 (ii)
ℓ crosses Λ at least Λ=P−2k times;
 (iii)
for each of the points \(p_{i}^{*}\wedge q^{*}\), replacing the crossing point by a contact point removes two crossings with ℓ.
Remark 28
Finally, contrarily to pseudotriangulations (k=1) and multitriangulations (convex position), the condition of avoiding (k+1)crossings does not hold for kpseudotriangulations in general:
4.3 Stars in Multipseudotriangulations
To complete our understanding of the primal of multipseudotriangulations, we need to generalize pseudotriangles of pseudotriangulations and kstars of ktriangulations: both pseudotriangles and kstars correspond to pseudolines of the covering pseudoline arrangement.
We keep the notations of the previous section: P is a point set in general position, Λ is a kpseudotriangulation of P ^{∗} and E is the primal set of edges of Λ.
Definition 30
Lemma 31
For any λ∈Λ, the star S(λ) is nonempty.
Proof
We have to prove that any pseudoline λ of Λ supports at least one contact point. If it is not the case, then λ is also a pseudoline of P ^{∗}, and all the P−1 crossing points of λ with P ^{∗}∖{λ} should be crossing points of λ with the arrangement Λ∖{λ}. This is impossible since Λ∖{λ}=P−2k−1. □
Similarly to the case of ktriangulations of the ngon, we say that an edge [p,q] of E is a krelevant (resp. kboundary, resp. kirrelevant) edge if there remain strictly more than (resp. exactly, resp. strictly less than) k−1 points of P on each side (resp. one side) of the line (pq). In other words, p ^{∗}∧q ^{∗} is located in the kkernel (resp. in the intersection of the kth level and the kkernel, resp. in the first k levels) of the pseudoline arrangement P ^{∗}. Thus, the edge [p,q] is contained in 2 (resp. 1, resp. 0) stars of Λ.
We call kdepth of a point q the number δ ^{ k }(q) of kboundary edges of P crossed by any (generic continuous) path from q to the external face, counted positively when passing from the “big” side (the one containing at least k vertices of P) to the “small side” (the one containing k−1 vertices of P), and negatively otherwise (see Fig. 19(b)). That this number is independent from the path can be seen by mutation. For example, δ ^{1}(q) is 1 if q is in the convex hull of P and 0 otherwise.
Proposition 32
 (i)
If e is kirrelevant, it is not contained in any star of Λ, and we do not change the winding numbers of the stars of Λ.
 (ii)
If e is a kboundary edge, and if we cross it positively, we increase the winding number of the star S of Λ containing e; if we cross e negatively, we decrease the winding number of S.
 (iii)
If e is krelevant, then we decrease the winding number of one star of Λ containing e and increase the winding number of the other star of Λ containing e.
Proof of Proposition 32
 (i)
If τ _{ λ }(q) denotes the number of intersection points between q ^{∗} and λ (that is, the number of tangents to S(λ) passing through q), then σ _{ λ }(q)=(τ _{ λ }(q)−1)/2.
 (ii)
If γ ^{ k }(q) denotes the number of intersection points between q ^{∗} and the first k levels of P ^{∗}, then δ ^{ k }(q)=k−γ ^{ k }(q)/2.
Remark 33
As a consequence of Proposition 32, we see that the kdepth of any point q in any point set P is always nonnegative (as a sum of nonnegative numbers). It is interesting to notice that Welzl proved in [44] that this nonnegativity property is actually equivalent to the Lower Bound Theorem for ddimensional polytopes with d+3 vertices.
Proposition 34
The number of corners of a star S(λ) of a kpseudotriangulation of P ^{∗} is odd and between 2k+1 and 2(k−1)P+2k+1.
Proof
 (i)
either on opposite sides and then we are not changing c;
 (ii)
or on the same side and we are adding to c either 0 (if \(\bar{w}\) is also on the same side) or 2 (if \(\bar{w}\) is on the opposite side).
To prove the lower bound, we use our witness method. We perturb λ a little bit to obtain a pseudoline μ that passes on the opposite side of each contact point (this is possible since c is odd). This pseudoline μ crosses λ between each pair of opposite contact points and crosses the other pseudolines of Λ exactly as λ does. Thus, μ crosses Λ exactly Λ−1+c times. But since μ is a pseudoline, it has to cross all the pseudolines of P ^{∗} at least once. Thus, P≤Λ−1+c=P−2k−1+c and c≥2k+1.
5 Iterated Multipseudotriangulations
By definition, a kpseudotriangulation of an mpseudotriangulation of a pseudoline arrangement L is a (k+m)pseudotriangulation of L. In this section, we study these iterated sequences of multipseudotriangulations. In particular, we compare multipseudotriangulations with iterated sequences of 1pseudotriangulations.
5.1 Definition and Examples
Let L be a pseudoline arrangement. An iterated multipseudotriangulation of L is a sequence Λ _{1},…,Λ _{ r } of pseudoline arrangements such that Λ _{ i } is a multipseudotriangulation of Λ _{ i−1} for all i (by convention, Λ _{0}=L). We call signature of Λ _{1},…,Λ _{ r } the sequence k _{1}<⋯<k _{ r } of integers such that Λ _{ i } is a k _{ i }pseudotriangulation of L for all i. Observe that the assumption that contact points of a pseudoline arrangement L should be contact points of any multipseudotriangulation of L is natural in this setting: iterated multipseudotriangulations correspond to decreasing sequences of sets of crossing points.
A decomposition of a multipseudotriangulation Λ of a pseudoline arrangement L is an iterated multipseudotriangulation Λ _{1},…,Λ _{ r } of L such that Λ _{ r }=Λ and r>1. We say that Λ is decomposable if such a decomposition exists, and irreducible otherwise. The decomposition is complete if its signature is 1,2,…,r.
It is tempting to believe that all multipseudotriangulations are completely decomposable. This would allow to focus only on pseudotriangulations. However, we start by showing that not even all multitriangulations are decomposable. The following example is due to Francisco Santos.
Example 35
(An irreducible 2triangulation of the 15gon)
 (i)
all the 2irrelevant and 2boundary edges of the 15gon, and
 (ii)
the five zigzags Z _{ a }={[3a,3a+6],[3a+6,3a+1],[3a+1,3a+5], [3a+5,3a+2]}, for a∈{0,1,2,3,4}.
Let us now prove that T ^{∗} is irreducible, that is, that T contains no triangulation. Observe first that the edge [0,6] cannot be an edge of a triangulation contained in T since none of the triangles 06i, i∈{7,…,14}, is contained in T. Thus, we are looking for a triangulation contained in T∖{[0,6]}. Repeating the argument successively for the edges [1,6], [1,5] and [2,5], we prove that the zigzag Z _{0} is disjoint from any triangulation contained in T. By symmetry, this proves the irreducibility of T ^{∗}.
5.2 Iterated Greedy Pseudotriangulations
Greedy multipseudotriangulations provide interesting examples of iteration of pseudotriangulations. Let L be a pseudoline arrangement, and χ be a cut of L.
Theorem 36
For any positive integers a and b, \(\varGamma _{\chi}^{a+b}(L)=\varGamma _{\chi}^{b}(\varGamma _{\chi}^{a}(L))\). Consequently, for any integer k, \(\varGamma _{\chi}^{k}(L)=\varGamma _{\chi}^{1}\circ \varGamma _{\chi}^{1}\circ\cdots\circ \varGamma _{\chi}^{1}(L)\), where \(\varGamma _{\chi}^{1}(.)\) is iterated k times.
Proof
Since χ is a cut of L, it is also a cut of \(\varGamma ^{a}_{\chi}(L)\) and thus \(\varGamma ^{b}_{\chi}(\varGamma ^{a}_{\chi}(L))\) is well defined. Observe also that we can assume that L has no contact point (otherwise, we can open them). Let n:=L and \(m :={n \choose2}\).
Let χ=χ _{0},…,χ _{ m }=χ be a backward sweep of L. For all i, let v _{ i } denote the vertex of L swept when passing from χ _{ i } to χ _{ i+1}, and i ^{□} denote the integer such that the pseudolines that cross at v _{ i } are the i ^{□}th and (i ^{□}+1)th pseudolines of L on χ _{ i }.

ρ _{ i+1} is obtained from ρ _{ i } by sorting its i ^{□}th and (i ^{□}+1)th entries;

if \(v_{i}\notin \varGamma ^{a}_{\chi}(L)\), then ω _{ i+1} is obtained from ω _{ i } by sorting its i ^{□}th and (i ^{□}+1)th entries; otherwise, ω _{ i+1}=ω _{ i }.
 (A)
all the inversions of ρ _{ i } are also inversions of σ _{ i }: ρ _{ i }(p)>ρ _{ i }(q) implies σ _{ i }(p)>σ _{ i }(q) for all 1≤p<q≤n; and
 (B)
ρ _{ i }=ω _{ i }.
 (1)
First case: σ _{ i }(i ^{□})<σ _{ i }(i ^{□}+1). Then σ _{ i+1}=σ _{ i } and \({v_{i}\in \varGamma ^{a}_{\chi}(L)}\). Thus, ω _{ i+1}=ω _{ i }. Furthermore, using property (A) at rank i, we know that ρ _{ i }(i ^{□})<ρ _{ i }(i ^{□}+1), and thus ρ _{ i+1}=ρ _{ i }. To summarize, σ _{ i+1}=σ _{ i }, ω _{ i+1}=ω _{ i }, and ρ _{ i+1}=ρ _{ i }, which trivially implies that properties (A) and (B) remain true.
 (2)Second case: σ _{ i }(i ^{□})>σ _{ i }(i ^{□}+1). Then σ _{ i+1} is obtained from σ _{ i } by exchanging the i ^{□}th and (i ^{□}+1)th entries, and \(v_{i}\notin \varGamma ^{a}_{\chi}(L)\). Consequently, ρ _{ i+1} and ω _{ i+1} are both obtained from ρ _{ i } and ω _{ i }, respectively, by sorting their i ^{□}th and (i ^{□}+1)th entries. Thus, property (B) obviously remains true. As far as property (A) is concerned, the result is obvious if p and q are different from i ^{□} and i ^{□}+1. By symmetry, it suffices to prove that for any p<i ^{□}, ρ _{ i+1}(p)>ρ _{ i+1}(i ^{□}) implies σ _{ i+1}(p)>σ _{ i+1}(i ^{□}). We have to consider two subcases:
 (a)
First subcase: ρ _{ i }(i ^{□})<ρ _{ i }(i ^{□}+1). Then ρ _{ i+1}=ρ _{ i }. Thus, if p<i ^{□} is such that ρ _{ i+1}(p)>ρ _{ i+1}(i ^{□}), then we have ρ _{ i }(p)>ρ _{ i }(i ^{□}). Consequently, we obtain σ _{ i+1}(p)=σ _{ i }(p)>σ _{ i }(i ^{□})>σ _{ i }(i ^{□}+1)=σ _{ i+1}(i ^{□}).
 (b)
Second subcase: ρ _{ i }(i ^{□})>ρ _{ i }(i ^{□}+1). Then ρ _{ i+1} is obtained from ρ _{ i } by exchanging its i ^{□}th and (i ^{□}+1)th entries. If p<i ^{□} is such that ρ _{ i+1}(p)>ρ _{ i+1}(i ^{□}), then we have ρ _{ i }(p)>ρ _{ i }(i ^{□}+1). Consequently, we obtain σ _{ i+1}(p)=σ _{ i }(p)>σ _{ i }(i ^{□}+1)=σ _{ i+1}(i ^{□}).
 (a)
5.3 Flips in Iterated Multipseudotriangulations
 (i)
If j=r, then \(\varLambda _{1},\dots,\varLambda _{i1},\varLambda '_{i},\dots ,\varLambda '_{r}\) is an iterated multipseudotriangulation of L. We say that it is obtained from Λ _{1},…,Λ _{ r } by a complete flip of v.
 (ii)If j<r, and w _{ i }=w _{ j } is a contact point of Λ _{ j+1}, thenis an iterated multipseudotriangulation of L. We say that it is obtained from Λ _{1},…,Λ _{ r } by a partial flip of v.$$\varLambda _1,\dots,\varLambda _{i1},\varLambda '_i,\dots,\varLambda '_j,\varLambda _{j+1},\dots,\varLambda _r$$
 (iii)
If j<r, and w _{ i }=w _{ j } is a crossing point in Λ _{ j+1}, then we cannot flip v in Λ _{ i } maintaining an iterated multipseudotriangulation of L.
 (i)
points u′ and u″ coincide. Thus we can flip simultaneously point u in (b) and (c) (complete flip);
 (ii)
points v′ is different from v″ but is a contact point in (c). Thus, we can just flip v in (b), without changing (c) and we preserve an iterated pseudotriangulation (partial flip);
 (iii)
point w′ is a crossing point in (c), different from w″. Thus, we cannot flip w in (b) maintaining an iterated pseudotriangulation.
Let \(G^{k_{1},\dots,k_{r}}(L)\) be the graph whose vertices are the iterated multipseudotriangulations of L with signature k _{1}<…<k _{ r }, and whose edges are the pairs of iterated multipseudotriangulations linked by a (complete or partial) flip.
Theorem 37
The graph of flips \(G^{k_{1},\dots,k_{r}}(L)\) is connected.
To prove this proposition, we need the following lemma:
Lemma 38
Any intersection point v in the kkernel of a pseudoline arrangement is a contact point in a kpseudotriangulation of it.
Proof
The result holds when k=1. We obtain the general case by iteration. □
Proof of Theorem 37
We prove the result by induction on r (L is fixed). When r=1, we already know that the flip graph is connected. Now, let A _{−} and A _{+} be two iterated multipseudotriangulations of L with signature k _{1}<…<k _{ r } that we want to join by flips. Let B _{−} and B _{+} be iterated multipseudotriangulations of L with signature k _{1}<…<k _{ r−1}, and Λ _{−} and Λ _{+} be k _{ r }pseudotriangulations of L such that A _{−}=B _{−},Λ _{−} and A _{+}=B _{+},Λ _{+}.
Merging all these paths, we obtain a global path from A _{−} to A _{+}: we transform A _{−} into C _{1} via a path of complete flips; then C _{1} into D _{1} by the partial flip of v _{1}; then D _{1} into C _{2} via a path of complete flips; then C _{2} into D _{2} by the partial flip of v _{2}; and so on. □
6 Further Topics
 (1)
The first one concerns the connection between the greedy pseudotriangulation of a point set and its horizon trees.
 (2)
The second one extends to arrangements of double pseudolines the definition and properties of multipseudotriangulations.
6.1 Greedy Multipseudotriangulations and Horizon Graphs
 (1)
the unique source of the graph of increasing flips;
 (2)
a greedy choice of crossing points given by a sorting network;
 (3)
a greedy choice of contact points;
 (4)
an iteration of greedy 1pseudotriangulations.
Let L be a pseudoline arrangement, and χ be a cut of L. We index by ℓ _{1},…,ℓ _{ n } the pseudolines of L in the order in which they cross χ (it is well defined, up to a complete inversion).
We define the kupper χhorizon set of L to be the set \(\mathbb {U}^{k}_{\chi}(L)\) of crossing points ℓ _{ α }∧ℓ _{ β }, with 1≤α<β≤n, such that there is no γ _{1},…,γ _{ k } satisfying α<γ _{1}<⋯<γ _{ k } and \(\ell_{\alpha}\wedge\ell_{\gamma_{i}}\preccurlyeq _{\chi}\ell_{\alpha}\wedge\ell_{\beta}\) for all i∈[k]. In other words, on each pseudoline ℓ _{ α } of L, the set \(\mathbb {U}^{k}_{\chi}(L)\) consists of the smallest k crossing points of the form ℓ _{ α }∧ℓ _{ β }, with α<β.
Similarly, define the klower χhorizon set of L to be the set \(\mathbb {L}^{k}_{\chi}(L)\) of crossing points ℓ _{ α }∧ℓ _{ β }, with 1≤α<β≤n, such that there is no δ _{1},…,δ _{ k } satisfying δ _{1}<⋯<δ _{ k }<β and \({\ell _{\beta}\wedge\ell_{\delta_{i}}\preccurlyeq _{\chi}\ell_{\alpha}\wedge\ell_{\beta}}\) for all i∈[k]. On each pseudoline ℓ _{ β } of L, the set \(\mathbb {L}^{k}_{\chi}(L)\) consists of the smallest k crossing points of the form ℓ _{ α }∧ℓ _{ β }, with α<β.
 (i)
α<γ _{1}<…<γ _{ k }, δ _{1}<…<δ _{ k }<β, and δ _{ k }<γ _{1}; and
 (ii)
\(\ell_{\alpha}\wedge\ell_{\gamma_{i}}\preccurlyeq _{\chi}\ell_{\alpha}\wedge\ell _{\beta}\) and \(\ell_{\beta}\wedge\ell_{\delta_{i}}\preccurlyeq _{\chi}\ell_{\alpha}\wedge \ell_{\beta}\) for all i∈[k].
Obviously, the sets \(\mathbb {U}^{k}_{\chi}(L)\) and \(\mathbb {L}^{k}_{\chi}(L)\) are both contained in \(\mathbb {G}^{k}_{\chi}(L)\).
Example 39
 (△)

crossing points of the kupper χhorizon set \(\mathbb {U}_{\chi}^{k}(L)\) are represented by up triangles △;
 ( Open image in new window )

crossing points of the klower χhorizon set \(\mathbb {L}_{\chi}^{k}(L)\) are represented by down triangles Open image in new window ;
 ( Open image in new window )

crossing points in both \(\mathbb {U}_{\chi}^{k}(L)\) and \(\mathbb {L}_{\chi}^{k}(L)\) are represented by up and down triangles Open image in new window ;
 (□)

crossing points of \(\mathbb {G}_{\chi}^{k}(L)\) but neither in \(\mathbb {U}_{\chi}^{k}(L)\), nor in \(\mathbb {L}_{\chi}^{k}(L)\) are represented by squares □.
Example 40
 (i)
\(\mathbb {U}^{k}_{\chi}(C_{n}^{*})=\{\ell_{\alpha}\wedge\ell_{\beta}\mid 1\le\alpha \le n\text{ and }\alpha<\beta\le\alpha+k\}\);
 (ii)
\(\mathbb {L}^{k}_{\chi}(C_{n}^{*})=\{\ell_{\alpha}\wedge\ell_{\beta}\mid 1\le\alpha \le k\text{ and }\alpha< j\le n\}\); and
 (iii)
\(\mathbb {G}^{k}_{\chi}(C_{n}^{*})=\mathbb {U}^{k}_{\chi}(\mathcal {C}_{n})\cup \mathbb {L}^{k}_{\chi}(\mathcal {C}_{n})\).
Theorem 41 extends this observation to all pseudoline arrangements, using convex position as a starting point for a proof by mutation.
Theorem 41
For any pseudoline arrangement L with no contact point, and any cut χ of L, the sets \(V(\varGamma _{\chi}^{k}(L))\) and \(\mathbb {G}^{k}_{\chi}(L)\) coincide.
If P is a point set of a topological plane, mutating an empty triangle p ^{∗} q ^{∗} r ^{∗} of P ^{∗} by sweeping the vertex q ^{∗}∧r ^{∗} with the pseudoline p ^{∗} corresponds in the primal to moving p a little bit such that only the orientation of the triangle pqr changes.
The graph of mutations on pseudoline arrangements is known to be connected: any two pseudoline arrangements (with no contact points and the same number of pseudolines) are homotopic via a finite sequence of mutations (followed by a homeomorphism). In fact, one can even avoid mutations of triangles that cross a given cut of L:
Proposition 42
Let L and L′ be two pseudoline arrangements of \(\mathcal {M}\) (with no contact points and the same number of pseudolines) and χ be a cut of both of L and L′. There is a finite sequence of mutations of triangles disjoint from χ that transforms L into L′.
Proof
We prove that any arrangement L of n pseudolines can be transformed into the arrangement \(C_{n}^{*}\) of n pseudolines in convex position (see Fig. 25).
Let ℓ _{1},…,ℓ _{ n } denote the pseudolines of L (ordered by their crossings with χ). Let Δ denote the triangle formed by χ, ℓ _{1} and ℓ _{2}. If there is a vertex of the arrangement L∖{ℓ _{1},ℓ _{2}} inside Δ, then there is a triangle of the arrangement L inside Δ and adjacent to ℓ _{1} or ℓ _{2}. Mutating this triangle reduces the number of vertices of L∖{ℓ _{1},ℓ _{2}} inside Δ such that after some mutations, there is no more vertex inside Δ. If Δ is intersected by pseudolines of L∖{ℓ _{1},ℓ _{2}}, then there is a triangle inside Δ formed by ℓ _{1}, ℓ _{2} and one of these intersecting pseudolines (the one closest to ℓ _{1}∧ℓ _{2}). Mutating this triangle reduces the number of pseudolines intersecting Δ. Thus, after some mutations, Δ is a triangle of the arrangement L.
Repeating these arguments, we prove that for all i∈{2,…,n−1} and after some mutations, ℓ _{ i }, ℓ _{1}, ℓ _{ i+1} and χ delimit a face of the arrangement L. Thus, one of the two topological disk delimited by χ and ℓ _{1} contains no more vertex of L, and the proof is then straightforward by induction. □
Lemma 43
Proof
By symmetry, it is enough to prove the first line of each of the four points of the lemma.
Properties of point (i) directly come from the definitions. For example, all the assertions of the first line are false if and only if there exist γ _{1},…,γ _{ k−1} with a<γ _{1}<⋯<γ _{ k−1} and, for all i∈[k−1], \(\ell_{a}\wedge\ell_{\gamma_{i}}\preccurlyeq _{\chi}C\) (or equivalently \(\ell_{a}\wedge\ell_{\gamma_{i}}\preccurlyeq _{\chi}E\)).
We derive point (ii) from the following observation: if γ>b and if ℓ _{ b }∧ℓ _{ γ }≼_{ χ } C, then γ>a and ℓ _{ a }∧ℓ _{ γ }≼_{ χ } B.
For point (iii), assume that \(A\notin \mathbb {U}_{\chi}^{k}(L)\) and \(C\notin \mathbb {L}_{\chi}^{k}(L)\). Then there exist γ _{1},…,γ _{ k } and δ _{1},…,δ _{ k } such that δ _{1}<⋯<δ _{ k }<b<γ _{1}<⋯<γ _{ k } and, for all i∈[k], \(\ell_{b}\wedge\ell_{\gamma_{i}}\preccurlyeq _{\chi}A\) (and therefore \(\ell_{a}\wedge \ell_{\gamma_{i}}\preccurlyeq _{\chi}C\)) and \(\ell_{b}\wedge\ell_{\delta_{i}}\preccurlyeq _{\chi}C\). Thus \(C\notin \mathbb {G}_{\chi}^{k}(L)\).
Finally, assume that \(C\in \mathbb {G}_{\chi}^{k}(L)\) and \(E\notin \mathbb {G}_{\chi}^{k}(L')\). Then there exist γ _{1},…,γ _{ k } and δ _{1},…,δ _{ k } such that a<γ _{1}<⋯<γ _{ k }, δ _{1}<⋯<δ _{ k }<c, δ _{ k }<γ _{1}, and for all i∈[k], \(\ell_{a}\wedge\ell_{\gamma _{i}}\preccurlyeq _{\chi}E\) and \(\ell_{c}\wedge\ell_{\delta_{i}}\preccurlyeq _{\chi}E\). Since \(C\in \mathbb {G}_{\chi}^{k}(L)\), we have δ _{ k }>b. Thus b<γ _{1}<⋯<γ _{ k } and for all i∈[k], \(\ell_{b}\wedge\ell_{\gamma_{i}}\preccurlyeq _{\chi}A\) and \(\ell_{c}\wedge\ell_{\delta_{i}}\preccurlyeq _{\chi}A\). This implies that \(A\notin \mathbb {G}_{C}^{k}(L)\). □
We are now ready to establish the proof of Theorem 41:
Proof of Theorem 41
The proof works by mutation. We already observed the result when the pseudoline arrangement is in convex position (see Example 40 and Fig. 25). Proposition 42 ensures that any pseudoline arrangement can be reached from this convex configuration by mutations of triangles not intersecting χ. Thus, it is sufficient to prove that such a mutation preserves the property.
Assume that L is a pseudoline arrangement and χ is a cut of L, for which the result holds. Let ∇ be a triangle of L not intersecting χ. Let L′ denote the pseudoline arrangement obtained from L by mutating the triangle ∇ into the inverted triangle Δ. Let A,B,C and D,E,F denote the vertices of ∇ and Δ as indicated in Fig. 27.
 (i)
either \(\{A,B,C\}\subset \mathbb {G}_{\chi}^{k}(L)\) and \(\{D,E,F\}\subset \mathbb {G}_{\chi}^{k}(L')\);
 (ii)
or \(\{A,B,C\}\cap \mathbb {G}_{\chi}^{k}(L)=\{A,C\}\) and \(\{D,E,F\}\cap \mathbb {G}_{\chi}^{k}(L')=\{D,E\}\);
 (iii)
or \(\{A,B,C\}\cap \mathbb {G}_{\chi}^{k}(L)=\{B,C\}\) and \(\{D,E,F\}\cap \mathbb {G}_{\chi}^{k}(L')=\{D,F\}\);
 (iv)
or \(\{A,B,C\}\cap \mathbb {G}_{\chi}^{k}(L)=\{A\}\) and \(\{D,E,F\}\cap \mathbb {G}_{\chi}^{k}(L')=\{D\}\);
 (v)
or \(\{A,B,C\}\cap \mathbb {G}_{\chi}^{k}(L)=\{C\}\) and \(\{D,E,F\}\cap \mathbb {G}_{\chi}^{k}(L')=\{F\}\);
 (vi)
or \(\{A,B,C\}\cap \mathbb {G}_{\chi}^{k}(L)=\emptyset\) and \(\{D,E,F\}\cap \mathbb {G}_{\chi}^{k}(L')=\emptyset\).
Choosing a cut χ of P ^{∗} corresponding to the point at infinity (−∞,0) makes coincide primal and dual definitions of horizon sets: we have \(\mathbb {U}^{1}_{\chi}(P^{*})=\mathbb {U}(P)^{*}\) and \(\mathbb {L}^{1}_{\chi}(P^{*})=\mathbb {L}(P)^{*}\).
In [31], Pocchiola observed that the set \(\mathbb {U}(P)\cup \mathbb {L}(P)\) of edges can be completed into a pseudotriangulation of P just by adding the sources of the faces of P ^{∗} intersected by the cut χ. The obtained pseudotriangulation is our χgreedy 1pseudotriangulation \(\varGamma ^{1}_{\chi}(P^{*})\).
6.2 Multipseudotriangulations of Double Pseudoline Arrangements
In this section, we deal with double pseudoline arrangements, i.e. duals of sets of disjoint convex bodies. Definitions and properties of multipseudotriangulations naturally extend to these objects.
6.2.1 Definitions
 (i)
either contractible (homotopic to a point);
 (ii)
or non separating, or equivalently homotopic to a generator of the fundamental group of \(\mathcal {M}\): it is a pseudoline;
 (iii)or separating and noncontractible, or equivalently homotopic to the double of a generator of the fundamental group of \(\mathcal {M}\): it is called a double pseudoline (see Fig. 30(a)).
The complement of a double pseudoline ℓ has two connected components: the bounded one is a Möbius strip M _{ ℓ } and the unbounded one is an open cylinder C _{ ℓ } (see Fig. 30(a)).
Definition 44
[17]
A double pseudoline arrangement is a finite set of double pseudolines such that any two of them have exactly four intersection points, cross transversally at these points, and induce a cell decomposition of the Möbius strip (i.e. the complement of their union is a union of topological disks, together with the external cell).
Given a set Q of disjoint convex bodies in the plane (or in any topological plane), its dual Q ^{∗}:={C ^{∗}∣C∈Q} is an arrangement of double pseudolines (see Figs. 31 and 32). Furthermore, as for pseudoline arrangements, any double pseudoline arrangement can be represented by (i.e. is the dual of) a set of disjoint convex bodies in a topological plane [17].
Definition 45
A kpseudotriangulation of a double pseudoline arrangement L is a pseudoline arrangement supported by the kkernel of L.
All the properties related to flips developed in Sect. 2 apply in this context. In the end of this section, we only revisit the properties of the primal of a multipseudotriangulation of a double pseudoline arrangement.
6.2.2 Elementary Properties
Let Q be a set disjoint convex bodies in general position in the plane and Q ^{∗} be its dual arrangement. Let Λ be a kpseudotriangulation of Q ^{∗}, V(Λ) denote all crossing points of Q ^{∗} that are not crossing points of Λ, and E denote the corresponding set of bitangents of Q. As in Sect. 4.2, we discuss the properties of the primal configuration E:
Lemma 46
The set E has 4Qk−Q−2k ^{2}−k edges.
Proof
We now discuss pointedness. For any convex body C of Q, we arbitrarily choose a point p _{ C } in the interior of C, and we consider the set X _{ C } of all segments between p _{ C } and a sharp boundary point of C. We denote by X:=⋃_{ C∈Q } X _{ C } the set of all these segments.
Lemma 47
The set E∪X cannot contain a kalternation.
Proof
 (i)
ℓ crosses Q ^{∗} exactly 2(Q∖{C})^{∗}+2k=2Q+2k−2 times;
 (ii)
ℓ crosses Λ at least Λ=2Q−2k times;
 (iii)
for each of the points p _{ i }∧q ^{∗}, replacing the crossing point by a contact point removes two crossings with ℓ.
6.2.3 Stars
 (i)
all bitangents τ between two convex bodies of Q such that τ is a contact point of λ; and
 (ii)
all convex arcs formed by the tangent points of the lines covered by λ with the convex bodies of Q.
Proposition 48
The number of corners of S(λ) is odd and between 2k−1 and 4kQ−2k−1.
Proof
In the case of double pseudoline arrangements, corners are even easier to characterize: a bitangent τ between two convex C and C′ of Q always defines two corners, one at each extremity. These corners are contained in one of the two stars adjacent to τ. Let λ be a pseudoline with a contact point at τ. In a neighborhood of τ, the pseudoline λ can be contained either in \(M_{C^{*}}\) or in \(C_{C^{*}}\). In the first case, the star S(λ) contains the corner formed by the bitangent τ and the convex C (or possibly, by the bitangent τ and another tangent to C); while in the second case, it does not. (The same observation holds for C′.)
 (i)
either v and w lie on opposite sides of λ; then exactly one of these contact points lies in \(M_{C^{*}}\), and S(λ) has one corner at C,
 (ii)
or v and w both lie on \(M_{C^{*}}\), and S(λ) has two corners at C,
 (iii)
or v and w both lie on \(C_{C^{*}}\), and S(λ) has no corners at C.
In order to get a lower bound on this number, we construct (as in the proof of Proposition 34) a witness pseudoline μ that crosses λ between each pair of opposite contact points and passes on the opposite side of each contact point. It crosses λ at most c times and Λ∖{λ} exactly Λ−1 times. Moreover, if α is a pseudoline and β is a double pseudoline of \(\mathcal {M}\), then either α is contained in M _{ β } and has no crossing with β, or α and β have an even number of crossings. Since μ is a pseudoline and can be contained in at most one Möbius strip \(M_{C}^{*}\) (for C∈Q), the number of crossings of μ with Q ^{∗} is at least 2(Q−1). Thus, we obtain the lower bound 2(Q−1)≤2Q−2k−1+c, i.e. c≥2k−1.
When k=1, we can even prove that all stars are pseudotriangles. Indeed, since any star has at least three corners, the upper bound calculus gives 2(3Q−3)≥c+3(2Q−3), i.e. c≤3.
Let us now give an analogue of Proposition 32. For any point q in the plane, we denote by η ^{ k }(q) the number of crossings between q ^{∗} and the support of Q ^{∗} minus its first k levels. Let δ ^{ k }(q):=η ^{ k }(q)/2−Q+k. For any λ∈Λ(U) and any point q in the plane, we still denote by σ _{ λ }(q) the winding number of S(λ) around q.
Proposition 49
For any point q of the plane δ ^{ k }(q)=∑_{ λ∈Λ } σ _{ λ }(q).
Proof
When k=1, it is easy to see that δ ^{1}(q) is 1 if q is inside the free space of the convex hull of Q (i.e. in the convex hull of Q, but not in Q), and 0 otherwise. Remember that a pseudotriangulation of Q is a pointed set of bitangents that decomposes the free space of the convex hull of Q into pseudotriangles [33]. Propositions 48 and 49 provide, when k=1, the following analogue of Theorem 16:
Theorem 50
 (i)
The dual arrangement T ^{∗}:={Δ^{∗}∣Δ pseudotriangle of T} of a pseudotriangulation T of Q is a 1pseudotriangulation of Q ^{∗}.
 (ii)The primal set of edgesof a 1pseudotriangulation Λ of Q ^{∗} is a pseudotriangulation of Q.$$\quad E :=\bigl \{[p,q] \,\big \,p,q\in P, p^*\wedge q^*\text{ is\ not\ a\ crossing\ point\ of\ } \varLambda \bigr \}$$
 (1)
either comparing the degrees of the flip graphs as in our first proof of Theorem 16;
 (2)
or checking that all forbidden configurations of the primal (two crossing bitangents, a nonpointed sharp vertex, a nonfree bitangent) may not appear in the dual, as in our second proof of Theorem 16.
For a point q outside ∇Q, the interpretation of δ ^{ k }(q) is similar to the case of points. We call level of a bitangent τ the level of the corresponding crossing point in Q ^{∗}. Given a point q outside ∇Q, the number δ ^{ k }(q) is the number of bitangents of level k crossed by any (generic continuous) path from q to the external face (in the complement of ∇Q), counted positively when passing from the “big” side to the “small side”, and negatively otherwise.
7 Open Questions
We finish by a short presentation of some open questions that have arisen out of this work. Since the submission of this paper, some of these questions were (at least partially) answered in [29, 40, 43]. We have decided to keep these questions in the discussion and to refer to the relevant articles in side remarks.
Primal of a Multipseudotriangulation
When k=1 or in the case of convex position, primals of kpseudotriangulations are characterized by simple noncrossing and pointedness conditions. For general k and general position, we know that the primal of a kpseudotriangulation is kalternationfree (Lemma 27), but there exist kpseudotriangulations containing (k+1)crossings as well as (k+1)crossingfree kalternationfree sets of edges not contained in kpseudotriangulations (see Fig. 18). Thus, we still miss a simple condition to characterize multipseudotriangulations of points in general position:
Question 51
Characterize primals of multipseudotriangulations.
Question 52
Does every point set in general position have a kpseudotriangulation with only “simple” stars (resp. only vertices with “little” degree)?
In the previous question, “simple” may be interpreted either as “with exactly 2k+1 corners” or as “with at most 2k+t edges” (for a minimal t). Similarly, “little” means smaller than a constant (as small as possible).
Diameter of the Graph of Flips
Question 53
What is the (asymptotic) diameter of the flip graph?
Polytopality
Question 54
Is \(\Delta(\mathcal {S})\) the boundary complex of a polytope?
 (a)Is the graph of flips on pseudotriangulations of a point set polytopal? Since [36] answers positively for Euclidean point sets, this question only remains open for nonstretchable arrangements. We have represented in Fig. 39 a pseudotriangulation of the smallest nonstretchable simple pseudoline arrangement (the nonPappus pseudoline arrangement). Since this arrangement is symmetric under the dihedral group D _{6}, it could be worked out with methods similar to those in [8].
 (b)
Is the graph of flips on ktriangulations of the ngon polytopal? Jonsson proved that the simplicial complex of (k+1)crossingfree sets of krelevant diagonals of the ngon is a combinatorial sphere [19]. However, except for little cases, the question of the polytopality of this complex remains open. We refer to [8, 28] and [26, Sect. 4.3] for a detailed discussion on this question.
Remark 55
Number of Multipseudotriangulations
In his paper [20], Jonsson proved that the number of ktriangulations of the ngon is equal to the determinant det(C _{ n−i−j })_{1≤i,j≤k } (where \(C_{m}=\frac{1}{m+1}{2m\choose m}\) denotes the mth Catalan number). This determinant also counts noncrossing ktuples of Dyck paths of semilength n−2k (see [28] and [26, Sect. 4.1] for a more detailed discussion). It raises the following question:
Question 56
Find an explicit bijection between Dyck multipaths and multitriangulations.
 (i)
All beams are x and ymonotone lattice paths.
 (ii)
The ith beam comes from the direction (−∞,k−1+i) and goes to the direction (k−1+i,+∞).
 (iii)
Each beam reflects exactly 2k+1 times, and thus, has k vertical segments (plus one vertical halfline) and k horizontal segments (plus one horizontal halfline).
 (iv)
The beams form a pseudoline arrangement: any two of them cross exactly once.
 (i)
the beam B _{ i } is (by duality) the set of all bisectors of S _{ i };
 (ii)
the mirrors which reflect B _{ i } are the edges of S _{ i }; and
 (iii)
the intersection of two beams B _{ i } and B _{ j } is the common bisector of S _{ i } and S _{ j }.
Remark 57
In their recent paper [40], Serrano and Stump also observe the correspondence between multitriangulations and beam arrangements (called “reduced pipe dream” in their paper). Starting from this correspondence, they provide an explicit bijection between multitriangulations and Dyck multipaths. We refer to [38, 40, 43] and the references therein.
Another interesting question concerning the number of multipseudotriangulations would be to determine what point sets give the maximum and minimum number of multipseudotriangulations. For example, when k=1, every point set in general position has at least as many pointed pseudotriangulations as the convex polygon with the same number of points [1].
Question 58
What point sets have the maximum and minimum number of multipseudotriangulations?
Decomposition of a kPseudotriangulation
We have seen in Sect. 5 that certain multipseudotriangulations can be decomposed into iterated multipseudotriangulations. Remember, however, that there exist irreducible multipseudotriangulations (Example 35).
Question 59
Characterize (completely) decomposable multipseudotriangulations.
The same question can be asked in a more algorithmical flavor:
Question 60
How can we decide algorithmically whether a kpseudotriangulation is decomposable?
Obviously, a bruteforce algorithm is not considered as a good solution. A good way to test efficiency of the answers to Questions 59 and 60 would be to prove/disprove that Example 35 is the minimal irreducible 2triangulation (or, in other words, that any 2triangulation of an ngon, with n≤14, contains a triangulation).
Remember also that when a multipseudotriangulation is decomposable, the graph of partial flips is not necessarily connected. In particular, finding all decompositions of a multipseudotriangulation cannot be achieved just by searching in the graph of partial flips.
Question 61
How can we enumerate all the decompositions of a multipseudotriangulation?
Computing a kPseudotriangulation
An initial pseudotriangulation of a set of n points can be computed in O(nlnn) time, using only the predicate of the chirotope. A similar result would be interesting for kpseudotriangulations:
Question 62
Compute an initial kpseudotriangulation of a given (double) pseudoline arrangement, using only its chirotope.
Notes
Acknowledgements
We thank Francisco Santos for fruitful discussions on the subject (especially for pointing out to us the counterexample of Fig. 22). During his visit at the École Normale Supérieure, Micha Sharir contributed a lot to the improvement of the paper. We thank Jakob Jonsson for pointed out an error in a previous formulation of Lemma 47. Finally, we are grateful to an anonymous reviewer for valuable feedback on the presentation of a preliminary version of this paper.
VP was partially supported by grant MTM200804699C0302 and MTM201122792 of the spanish Ministerio de Ciencia e Innovación, and by European Research Project ExploreMaps (ERC StG 208471). MP was partially supported by the TEOMATRO grant ANR10BLAN 0207.
References
 1.Aichholzer, O., Aurenhammer, F., Krasser, H., Speckmann, B.: Convexity minimizes pseudotriangulations. Comput. Geom. 28(1), 3–10 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
 2.Aichholzer, O., Orden, D., Santos, F., Speckmann, B.: On the number of pseudotriangulations of certain point sets. J. Comb. Theory, Ser. A 115(2), 254–278 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
 3.Angelier, P., Pocchiola, M.: A sum of squares theorem for visibility complexes and applications. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, the GoodmanPollack Festschrift. Algorithms Combin., vol. 25, pp. 79–139. Springer, Berlin (2003) Google Scholar
 4.Bereg, S.: Transforming pseudotriangulations. Inf. Process. Lett. 90(3), 141–145 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
 5.Bereg, S.: Enumerating pseudotriangulations in the plane. Comput. Geom. 30(3), 207–222 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
 6.Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 46. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
 7.Brönnimann, H., Kettner, L., Pocchiola, M., Snoeyink, J.: Counting and enumerating pointed pseudotriangulations with the greedy flip algorithm. SIAM J. Comput. 36(3), 721–739 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
 8.Bokowski, J., Pilaud, V.: On symmetric realizations of the simplicial complex of 3crossingfree sets of diagonals of the octagon. In: Proc. 25th Canadian Conference on Computational Geometry (2009) Google Scholar
 9.Capoyleas, V., Pach, J.: A Turántype theorem on chords of a convex polygon. J. Comb. Theory, Ser. A 56(1), 9–15 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
 10.de Bruijn, N.G.: Sorting by means of swappings. Discrete Math. 9, 333–339 (1974) MathSciNetzbMATHCrossRefGoogle Scholar
 11.Dress, A.W.M., Klucznik, M., Koolen, J.H., Moulton, V.: Open image in new window: note on extremal combinatorics of cyclic split systems. Sém. Lothar. Combin. 47, Article B47b (2001), 17 pp. (electronic) Google Scholar
 12.Dress, A.W.M., Koolen, J.H., Moulton, V.: On line arrangements in the hyperbolic plane. Eur. J. Comb. 23(5), 549–557 (2002) MathSciNetCrossRefGoogle Scholar
 13.Goodman, J.E.: Pseudoline arrangements. In: Handbook of Discrete and Computational Geometry. CRC Press Ser. Discrete Math. Appl., pp. 83–109. CRC, Boca Raton (1997) Google Scholar
 14.Grünbaum, B.: Arrangements and Spreads. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 10. American Mathematical Society, Providence (1972) zbMATHGoogle Scholar
 15.Haas, R., Orden, D., Rote, G., Santos, F., Servatius, B., Servatius, H., Souvaine, D., Streinu, I., Whiteley, W.: Planar minimally rigid graphs and pseudotriangulations. Comput. Geom. 31(1–2), 31–61 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
 16.Habert, L., Pocchiola, M.: Computing pseudotriangulations via branched coverings. Preprint, arXiv:1102.0151. Abbreviated partial version in 20th European Workshop on Comput. Geom. (Seville, 2004), pp. 111–114 (2007)
 17.Habert, L., Pocchiola, M.: Arrangements of double pseudolines. Preprint, arXiv:1101.1022. Abbreviated version in 25th Annual ACM Symposium on Comput. Geom. (Aahrus, 2009), pp. 314–323 (2009)
 18.Hohlweg, C., Lange, C.E.M.C.: Realizations of the associahedron and cyclohedron. Discrete Comput. Geom. 37(4), 517–543 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
 19.Jonsson, J.: Generalized triangulations of the ngon. Unpublished manuscript. An abstract was included in Topological and Geometric Combinatorics, April 2003, Mathematisches Forschungsinstitut Oberwolfach, Report No. 16 (2003) Google Scholar
 20.Jonsson, J.: Generalized triangulations and diagonalfree subsets of stack polyominoes. J. Comb. Theory, Ser. A 112(1), 117–142 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
 21.Kettner, L., Kirkpatrick, D., Mantler, A., Snoeyink, J., Speckmann, B., Takeuchi, F.: Tight degree bounds for pseudotriangulations of points. Comput. Geom. 25, 3–12 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
 22.Knutson, A., Miller, E.: Subword complexes in Coxeter groups. Adv. Math. 184(1), 161–176 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
 23.Knuth, D.E.: The Art of Computer Programming, vol. 3. AddisonWesley, Reading (1973). Sorting and Searching, AddisonWesley Series in Computer Science and Information Processing Google Scholar
 24.Knuth, D.E.: Axioms and Hulls. Lecture Notes in Computer Science, vol. 606. Springer, Berlin (1992) zbMATHCrossRefGoogle Scholar
 25.Nakamigawa, T.: A generalization of diagonal flips in a convex polygon. Theor. Comput. Sci. 235(2), 271–282 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
 26.Pilaud, V.: Multitriangulations, pseudotriangulations and some problems of realization of polytopes. Ph.D. thesis, Université Paris 7 & Universidad de Cantabria (2010) Google Scholar
 27.Pilaud, V., Pocchiola, M.: Multipseudotriangulations. In: 25th European Workshop on Comput. Geom., Bruxelles (2009). Extended abstract Google Scholar
 28.Pilaud, V., Santos, F.: Multitriangulations as complexes of star polygons. Discrete Comput. Geom. 41(2), 284–317 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
 29.Pilaud, V., Santos, F.: The brick polytope of a sorting network. Eur. J. Comb. 33(4), 632–662 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
 30.Pilaud, V., Stump, C.: Brick polytopes of spherical subword complexes: A new approach to generalized associahedra. Preprint, arXiv:1111.3349 (2011)
 31.Pocchiola, M.: Horizon trees versus pseudotriangulations. In: 13th European Workshop on Comput. Geom., Würzburg (1997) Google Scholar
 32.Pocchiola, M., Vegter, G.: Order types and visibility types of configurations of disjoint convex plane sets. Technical report, LIENS, Paris (1994) Google Scholar
 33.Pocchiola, M., Vegter, G.: Pseudotriangulations: theory and applications. In: 12th Annual ACM Symposium on Comput. Geom., Philadelphia, pp. 291–300 (1996) Google Scholar
 34.Pocchiola, M., Vegter, G.: Topologically sweeping visibility complexes via pseudotriangulations. Discrete Comput. Geom. 16(4), 419–453 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
 35.Pocchiola, M., Vegter, G.: The visibility complex. Int. J. Comput. Geom. Appl. 6(3), 279–308 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
 36.Rote, G., Santos, F., Streinu, I.: Expansive motions and the polytope of pointed pseudotriangulations. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, The GoodmanPollack Festschrift. Algorithms Combin., vol. 25, pp. 699–736. Springer, Berlin (2003) CrossRefGoogle Scholar
 37.Rote, G., Santos, F., Streinu, I.: Pseudotriangulations—a survey. In: Surveys on Discrete and Computational Geometry. Contemp. Math., vol. 453, pp. 343–410. Am. Math. Soc., Providence (2008) CrossRefGoogle Scholar
 38.Rubey, M.: Maximal 0–1 fillings of moon polyominoes with restricted chainlengths and RCgraphs. Preprint, arXiv:1009.3919 (2010)
 39.Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R., Stroppel, M.: Compact Projective Planes. de Gruyter Expositions in Mathematics, vol. 21. de Gruyter, Berlin (1995) CrossRefGoogle Scholar
 40.Serrano, L., Stump, C.: Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials. Electron. J. Comb. 19 (2012) Google Scholar
 41.Sleator, D.D., Tarjan, R.E., Thurston, W.P.: Rotation distance, triangulations, and hyperbolic geometry. J. Am. Math. Soc. 1(3), 647–681 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
 42.Streinu, I.: Pseudotriangulations, rigidity and motion planning. Discrete Comput. Geom. 34(4), 587–635 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
 43.Stump, C.: A new perspective on ktriangulations. J. Comb. Theory, Ser. A 118(6), 1794–1800 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
 44.Welzl, E.: Entering and leaving jfacets. Discrete Comput. Geom. 25(3), 351–364 (2001) MathSciNetzbMATHCrossRefGoogle Scholar