23 research outputs found

### Graph properties of graph associahedra

A graph associahedron is a simple polytope whose face lattice encodes the
nested structure of the connected subgraphs of a given graph. In this paper, we
study certain graph properties of the 1-skeleta of graph associahedra, such as
their diameter and their Hamiltonicity. Our results extend known results for
the classical associahedra (path associahedra) and permutahedra (complete graph
associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction

### Geometric realizations of the accordion complex of a dissection

Consider $2n$ points on the unit circle and a reference dissection
$\mathrm{D}_\circ$ of the convex hull of the odd points. The accordion complex
of $\mathrm{D}_\circ$ is the simplicial complex of non-crossing subsets of the
diagonals with even endpoints that cross a connected subset of diagonals of
$\mathrm{D}_\circ$. In particular, this complex is an associahedron when
$\mathrm{D}_\circ$ is a triangulation and a Stokes complex when
$\mathrm{D}_\circ$ is a quadrangulation. In this paper, we provide geometric
realizations (by polytopes and fans) of the accordion complex of any reference
dissection $\mathrm{D}_\circ$, generalizing known constructions arising from
cluster algebras.Comment: 25 pages, 10 figures; Version 3: minor correction

### Compatibility fans for graphical nested complexes

Graph associahedra are natural generalizations of the classical associahedra.
They provide polytopal realizations of the nested complex of a graph $G$,
defined as the simplicial complex whose vertices are the tubes (i.e. connected
induced subgraphs) of $G$ and whose faces are the tubings (i.e. collections of
pairwise nested or non-adjacent tubes) of $G$. The constructions of M. Carr and
S. Devadoss, of A. Postnikov, and of A. Zelevinsky for graph associahedra are
all based on the nested fan which coarsens the normal fan of the permutahedron.
In view of the combinatorial and geometric variety of simplicial fan
realizations of the classical associahedra, it is tempting to search for
alternative fans realizing graphical nested complexes.
Motivated by the analogy between finite type cluster complexes and graphical
nested complexes, we transpose in this paper S. Fomin and A. Zelevinsky's
construction of compatibility fans from the former to the latter setting. For
this, we define a compatibility degree between two tubes of a graph $G$. Our
main result asserts that the compatibility vectors of all tubes of $G$ with
respect to an arbitrary maximal tubing on $G$ support a complete simplicial fan
realizing the nested complex of $G$. In particular, when the graph $G$ is
reduced to a path, our compatibility degree lies in $\{-1,0,1\}$ and we recover
F. Santos' Catalan many simplicial fan realizations of the associahedron.Comment: 51 pages, 30 figures; Version 3: corrected proof of Theorem 2

### Compatibility fans realizing graphical nested complexes

Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra

### GÃ©nÃ©ralisations gÃ©omÃ©triques et combinatoires de l'associaÃ¨dre

The associahedron is at the interface between several mathematical fields. Combinatorially, it is the simplicial complex of dissections of a convex polygon (sets of mutually noncrossing diagonals). Geometrically, it is a polytope whose vertices and edges encode the dual graph of the complex of dissections. Finally the associahedron describes the combinatorial structure defining a presentation by generators and relations of certain algebras, called ``cluster algebras''. Because of its ubiquity, we regularly come up with new families generalizing this object. However there often are only few interactions between them. From our perspective, there are two main issues when studying them: looking for relations on the basis of known properties of the associahedron; and, for each, looking for combinatorial, geometric and algebraic frameworks in the same spirit.In this thesis, we deal with the link between combinatorics and geometry for some of these generalizations: graph associahedra, subword complexes and accordion complexes. We follow a guidelight consisting in adapting, to these three families, a method for constructing associahedra as fans (sets of polyhedral cones), called the d-vector method and coming from cluster algebra theory. More generally, our main concern is to realize, that is geometrically embed in a vector space, abstract complexes. We obtain three new families of generalizations, and a fourth conjectural one whose first instances already constitute significant advances.Finally in addition to the geometric results, we prove combinatorial properties specific to each encountered simplicial complex.L'associaÃ¨dre se situe Ã l'interface de plusieurs domaines mathÃ©matiques. Combinatoirement, il s'agit du complexe simplicial des dissections d'un polygone convexe (ensembles de diagonales ne se croisant pas deux Ã deux). GÃ©omÃ©triquement, il s'agit d'un polytope dont les sommets et les arÃªtes encodent le graphe dual du complexe des dissections. Enfin l'associaÃ¨dre dÃ©crit la structure combinatoire qui dÃ©finit la prÃ©sentation par gÃ©nÃ©rateurs et relations de certaines algÃ¨bres, dites >. Du fait de son omniprÃ©sence, de nouvelles familles gÃ©nÃ©ralisant cet objet sont rÃ©guliÃ¨rement dÃ©couvertes. Cependant elles n'ont souvent que de faibles interactions. Leurs Ã©tudes respectives prÃ©sentent de notre point de vue deux enjeux majeurs : chercher Ã les relier en se basant sur les propriÃ©tÃ©s connues de l'associaÃ¨dre ; et chercher pour chacune des cadres combinatoire, gÃ©omÃ©trique et algÃ©brique dans le mÃªme esprit.Dans cette thÃ¨se, nous traitons le lien entre combinatoire et gÃ©omÃ©trie pour certaines de ces gÃ©nÃ©ralisations : les associaÃ¨dres de graphes, les complexes de sous-mots et les complexes d'accordÃ©ons. Nous suivons un fil rouge consistant Ã adapter, Ã ces trois familles, une mÃ©thode de construction des associaÃ¨dres comme Ã©ventails (ensembles de cÃ´nes polyÃ©draux), dite mÃ©thode des d-vecteurs et issue de la thÃ©orie des algÃ¨bres amassÃ©es. De maniÃ¨re plus large, notre problÃ©matique principale consiste Ã rÃ©aliser, c'est-Ã -dire plonger gÃ©omÃ©triquement dans un espace vectoriel, des complexes abstraits. Nous obtenons trois familles de nouvelles rÃ©alisations, ainsi qu'une quatriÃ¨me encore conjecturale dont les premiÃ¨res instances constituent dÃ©jÃ des avancÃ©es significatives.Enfin, en sus des rÃ©sultats gÃ©omÃ©triques, nous dÃ©montrons des propriÃ©tÃ©s combinatoires spÃ©cifiques Ã chaque complexe simplicial abordÃ©

### Diameters and geodesic properties of generalizations of the associahedron

International audienceThe $n$-dimensional associahedron is a polytope whose vertices correspond to triangulations of a convex $(n + 3)$-gon and whose edges are flips between them. It was recently shown that the diameter of this polytope is $2n - 4$ as soon as $n > 9$. We study the diameters of the graphs of relevant generalizations of the associahedron: on the one hand the generalized associahedra arising from cluster algebras, and on the other hand the graph associahedra and nestohedra. Related to the diameter, we investigate the non-leaving-face property for these polytopes, which asserts that every geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both.Lâ€™associaÃ¨dre de dimension $n$ est un polytope dont les sommets correspondent aux triangulations dâ€™un $(n + 3)$-gone convexe et dont les arÃªtes sont les Ã©changes entre ces triangulations. Il a Ã©tÃ© rÃ©cemment dÃ©montrÃ© que le diamÃ¨tre de ce polytope est $2n - 4$ dÃ¨s que $n > 9$. Nous Ã©tudions les diamÃ¨tres des graphes de certaines gÃ©nÃ©ralisations de lâ€™associaÃ¨dre : dâ€™une part les associaÃ¨dres gÃ©nÃ©ralisÃ©s provenant des algÃ¨bres amassÃ©es, et dâ€™autre part les associaÃ¨dres de graphes et les nestoÃ¨dres. En lien avec le diamÃ¨tre, nous Ã©tudions si toutes les gÃ©odÃ©siques entre deux sommets de ces polytopes restent dans la plus petite face les contenant

### Compatibility fans realizing graphical nested complexes

Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra

### Compatibility fans realizing graphical nested complexes

International audienceGraph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra