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25
Multitriangulations, pseudotriangulations and primitive sorting networks
 Discrete Comput. Geom. (DOI
, 2012
"... 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 ..."
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Cited by 15 (10 self)
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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.
On the number of pseudotriangulations of certain point sets
 J. Combin. Theory Ser. A
, 2007
"... We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significant ..."
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Cited by 13 (3 self)
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We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de
BRICK POLYTOPES OF SPHERICAL SUBWORD COMPLEXES: A NEW APPROACH TO GENERALIZED ASSOCIAHEDRA
"... Abstract. We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polyt ..."
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Cited by 8 (0 self)
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Abstract. We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, and a Minkowski
DENOMINATOR VECTORS AND COMPATIBILITY DEGREES IN CLUSTER ALGEBRAS OF FINITE TYPE
"... Abstract. We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other ..."
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Cited by 4 (1 self)
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Abstract. We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the dvector of any noninitial cluster variable with respect to any initial cluster seed has nonnegative entries and is different from zero. 1.
Ellabelings and canonical spanning trees for subword complexes
, 2012
"... Abstract. We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facetridge graph of the subword complex, describe inductively these trees, and ..."
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Cited by 4 (2 self)
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Abstract. We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facetridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an ELlabeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar ELlabeling was recently studied by M. Kallipoliti and H. Mühle. Contents
The diameter of type D associahedra and the nonleavingface property
, 2014
"... Abstract. We prove that the graph diameter of the ndimensional associahedron of type D is precisely 2n − 2 for all n. Furthermore, we show that all type ABCD associahedra have the nonleavingface property, that is, any minimal path connecting two vertices in the graph of the polytope stays in the ..."
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Cited by 3 (2 self)
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Abstract. We prove that the graph diameter of the ndimensional associahedron of type D is precisely 2n − 2 for all n. Furthermore, we show that all type ABCD associahedra have the nonleavingface property, that is, any minimal path connecting two vertices in the graph of the polytope stays in the minimal face containing both. In contrast, we present relevant examples related to the associahedron that do not always satisfy this property.
Pointed Drawings of Planar Graphs
, 2008
"... We study the problem how to draw a planar graph such that every vertex is incident to an angle greater than π. In general a straightline embedding cannot guarantee this property. We present algorithms which construct such drawings with either tangentcontinuous biarcs or quadratic Bézier curves (par ..."
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Cited by 3 (1 self)
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We study the problem how to draw a planar graph such that every vertex is incident to an angle greater than π. In general a straightline embedding cannot guarantee this property. We present algorithms which construct such drawings with either tangentcontinuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane straightline embedding of the graph. Moreover, the graph can be embedded with circular arcs if the vertices can be placed arbitrarily. The topic is related to noncrossing drawings of multigraphs and vertex labeling.
Greedy flip trees in subword complexes
, 2012
"... Abstract. We describe a canonical spanning tree of the ridge graph of a subword complex on a finite Coxeter group. It is based on properties of greedy facets in subword complexes, defined and studied in this paper. Searching this tree yields an enumeration scheme for the facets of the subword comple ..."
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Cited by 2 (2 self)
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Abstract. We describe a canonical spanning tree of the ridge graph of a subword complex on a finite Coxeter group. It is based on properties of greedy facets in subword complexes, defined and studied in this paper. Searching this tree yields an enumeration scheme for the facets of the subword complex. This algorithm extends the greedy flip algorithm for pointed pseudotriangulations of points or convex bodies in the plane. 1.
Triangulations of Line Segment Sets in the Plane
"... Abstract. Given a set S of line segments in the plane, we introduce a new family of partitions of the convex hull of S called segment triangulations of S. The set of faces of such a triangulation is a maximal set of disjoint triangles that cut S at, and only at, their vertices. Surprisingly, several ..."
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Abstract. Given a set S of line segments in the plane, we introduce a new family of partitions of the convex hull of S called segment triangulations of S. The set of faces of such a triangulation is a maximal set of disjoint triangles that cut S at, and only at, their vertices. Surprisingly, several properties of point set triangulations extend to segment triangulations. Thus, the number of their faces is an invariant of S. In thesameway,ifS is in general position, there exists a unique segment triangulation of S whose faces are inscribable in circles whose interiors do not intersect S. This triangulation, called segment Delaunay triangulation, is dual to the segment Voronoi diagram. The main result of this paper is that the local optimality which characterizes point set Delaunay triangulations [10] extends to segment Delaunay triangulations. A similar result holds for segment triangulations with same topology as the Delaunay one. 1
Cluster algebras of type D: pseudotriangulations approach
"... We present a combinatorial model for cluster algebras of type Dn in terms of centrally symmetric pseudotriangulations of a regular 2ngon with a small disk in the centre. This model provides convenient and uniform interpretations for clusters, cluster variables and their exchange relations, as wel ..."
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We present a combinatorial model for cluster algebras of type Dn in terms of centrally symmetric pseudotriangulations of a regular 2ngon with a small disk in the centre. This model provides convenient and uniform interpretations for clusters, cluster variables and their exchange relations, as well as for quivers and their mutations. We also present a new combinatorial interpretation of cluster variables in terms of perfect matchings of a graph after deleting two of its vertices. This interpretation differs from known interpretations in the literature. Its main feature, in contrast with other interpretations, is that for a fixed initial cluster seed, one or two graphs serve for the computation of all cluster variables. Finally, we discuss applications of our model to polytopal realizations of type D associahedra and connections to subword complexes and ccluster complexes. 1