Graph Associahedra Research Articles

Colorful graph associahedra

Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.

Deformation cones of graph associahedra and nestohedra

We give the facet description of the deformation cones of graph associahedra and nestohedra, generalizing the classical parametrization of the family of deformed permutahedra by the cone of submodular functions. When the underlying building set is made of intervals, this leads in particular to the construction of kinematic nestohedra generalizing the kinematic associahedra that recently appeared in the theory of scattering amplitudes.

Diameter Estimates for Graph Associahedra

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of $\Theta(m)$ on the diameter of trivially perfect graph associahedra on $m$ edges. We consider the maximum diameter of associahedra of graphs on $n$ vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. We also prove that the maximum diameter of associahedra of graphs of pathwidth two is $\Theta (n\log n)$. Finally, we give the exact diameter of the associahedra of complete split and of unbalanced complete bipartite graphs.

Multivariate polynomials for generalized permutohedra

Using the notion of a Mahonian statistic on acyclic posets, we introduce a q-analogue of the h-polynomial of a simple generalized permutohedron. We focus primarily on the case of nestohedra and on explicit computations for many interesting examples, such as Sn-invariant nestohedra, graph associahedra, and Stanley-Pitman polytopes. For the usual (Stasheff) associahedron, our generalization yields an alternative q-analogue to the well-studied Narayana numbers.

Algebraic structures on graph associahedra

M. Carr and S. Devadoss introduced in ‘Coxeter complexes and graph associahedra’ [Topology Appl. 153 (1-2) (2006) 2155–2168] the notion of tubing on a finite simple graph Γ $\Gamma$ , in the context of configuration spaces on the Hilbert plane. To any finite simple graph Γ $\Gamma$ they associated a finite partially ordered set, whose elements are the tubings of Γ $\Gamma$ and whose geometric realization is a convex polytope K Γ ${\mathcal {K}}\Gamma$ , the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for the empty graph they obtained the standard simplexes. The goal of the present work is to give an algebraic description of graph associahedra. We introduce a substitution operation on tubings, which allows us to describe the set of faces of graph-associahedra as a free object, spanned by the set of all connected simple graphs, under operations given via connected subgraphs. The boundary maps of graph-associahedra define natural derivations in this context. Along the way, we introduce a topological interpretation of the graph tubings and our new operations. In the last section, we show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories.

Bounds on the Diameter of Graph Associahedra

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of search trees on G, defined recursively by a root r together with search trees on each of the connected components of G − r. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n - 6 for a large enough n. We give a tight bound of Θ(m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n).

Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra

The Malvenuto–Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto–Reutenauer algebra, both of which include the Loday–Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.

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.

Labelled Maximal Tubings on Paths and Graph Associahedra

We determine the triangulation of a cyclohedron compatible with the Tamari orderon its faces. We dene a name of a tubing on a path and a plumbing leading us to construct the dendriform algebra of the collection of maximal tubings on paths. Moreover, we give an operad structure of associahedra and a module structure of cyclohedra via tubings. We dene labelled maximal tubings on paths and give to an application of tubings in homological algebra.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.

Compatibility fans for graphical nested complexes

Graph associahedra generalize classical associahedra. They realize the nested complex of a graph G, i.e. the simplicial complex whose vertices are the tubes (connected induced subgraphs) of G and whose faces are the tubings (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 variety of fan realizations of associahedra, it is tempting to look for alternative fans realizing graphical nested complexes. Motivated by the analogy between finite type cluster complexes and graphical nested complexes, we transpose S. Fomin and A. Zelevinsky's compatibility fans from the former to the latter setting. We define a compatibility degree between two tubes of a graph G and show that the compatibility vectors of all tubes of G with respect to an arbitrary maximal tubing on G support a fan realizing the nested complex of G. When G is a path, we recover F. Santos' Catalan many realizations of the associahedron.

Pseudograph and its associated real toric manifold

Given a simple graph $G$, the graph associahedron $P_G$ is a convex polytope whose facets correspond to the connected induced subgraphs of $G$. Graph associahedra have been studied widely and are found in a broad range of subjects. Recently, S. Choi and H. Park computed the rational Betti numbers of the real toric variety corresponding to a graph associahedron under the canonical Delzant realization. In this paper, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss and Forcey, and then discuss how to compute the Poincare polynomial of the real toric variety corresponding to a pseudograph associahedron under the canonical Delzant realization.

Exponential formulas for models of complex reflection groups

In this paper we find exponential formulas for the Betti numbers of the De Concini–Procesi minimal wonderful models YG(r,p,n) associated to the complex reflection groups G(r,p,n). Our formulas are different from the ones already known in the literature: they are obtained by a new combinatorial encoding of the elements of a basis of the cohomology by means of set partitions with weights and exponents.We also point out that a similar combinatorial encoding can be used to describe the faces of the real spherical wonderful models of type An−1(=G(1,1,n)), Bn (=G(2,1,n)) and Dn(=G(2,2,n)). This provides exponential formulas for the f-vectors of the associated nestohedra: the Stasheff’s associahedra (in this case closed formulas are well known) and the graph associahedra of type Dn.

Toric graph associahedra and compactifications of $$M_{0,n}$$

To any graph G, one can associate a toric variety $$X(\mathcal {P}G)$$X(PG), obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of G. The polytopes of these toric varieties are the graph associahedra, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space $$X(\mathcal {P}{G})$$X(PG) is isomorphic to a Hassett compactification of $$M_{0,n}$$M0,n precisely when G is an iterated cone over a discrete set. This may be viewed as a generalization of the well-known fact that the Losev---Manin moduli space is isomorphic to the toric variety associated with the permutohedron.

Quilted strips, graph associahedra, andA∞n–modules

We consider moduli spaces of quilted strips with markings and their compactifications. Using the theory of moment maps of toric varieties we identify the compactified moduli spaces with certain graph associahedra. We demonstrate how these moduli spaces govern the combinatorics of A-infinity n-modules, which are natural generalizations of A-infinity modules (n=1) and bimodules (n=2).

Diameters and geodesic properties of generalizations of the associahedron

The $n$-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. L’associaèdre de dimension $n$ est un polytope dont les sommets correspondent aux triangulations d’un $(n + 3)$-gone convexe et dont les arêtes sont les échanges entre ces triangulations. Il a été récemment démontré que le diamètre de ce polytope est $2n - 4$ dès que $n > 9$. Nous étudions les diamètres des graphes de certaines généralisations de l’associaèdre : d’une part les associaèdres généralisés provenant des algèbres amassées, et d’autre part les associaèdres de graphes et les nestoèdres. En lien avec le diamètre, nous étudions si toutes les géodésiques entre deux sommets de ces polytopes restent dans la plus petite face les contenant.

Convex polytopes from nested posets

Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes. However, these poset associahedra fall in a different category altogether than generalized permutohedra.

Signed tree associahedra

An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. J.-L. Loday gave a particularly elegant realization of the associahedron, which was then generalized in two directions: on the one hand to obtain realizations of graph associahedra, and on the other hand to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and extend these two constructions to signed tree associahedra. Un associaèdre est un polytope dont les sommets correspondent aux triangulations d’un polygone convexe et dont les arêtes correspondent aux flips entre ces triangulations. J.-L. Loday a donné une construction particulièrement élégante de l’associaèdre qui a été généralisée dans deux directions : d’une part pour obtenir des réalisations des associaèdres de graphes, et d’autre part pour obtenir de multiples réalisations de l’associaèdre paramétrées par une suite de signes. L’objectif de ce travail est d’unifier et d’étendre ces constructions aux associaèdres d’arbres signés.

Toric symmetry of $\mathbb{CP}^3$

We exhaustively analyze the toric symmetries of $\mathbb{CP}^3$ and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov–Witten (GW) theory and Donaldson–Thomas (DT) theory. We identify all non-trivial toric symmetries. The induced nontrivial isomorphisms lift and provide new symmetries at the level of GW Theory and DT theory. The polytopes of the toric varieties in question include the permutohedron, the cyclohedron, the associahedron, and in fact all graph associahedra, among others.

Pseudograph associahedra

Given a simple graph G, the graph associahedron K G is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.