A Realization of Graph-Associahedra

The diagonal of the associahedra

This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.

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.

P-graph Associahedra and Hypercube Graph Associahedra

A graph associahedron is a polytope dual to a simplicial complex whose elements are induced connected subgraphs called tubes. Graph associahedra generalize permutahedra, associahedra, and cyclohedra, and therefore are of great interest to those who study Coxeter combinatorics. This thesis characterizes nested complexes of simplicial complexes, which we call $Δ$-nested complexes. From here, we can define P-nestohedra by truncating simple polyhedra, and in more specificity define P-graph associahedra, which are realized by repeated truncation of faces of simple polyhedra in accordance with tubes…

$\ u$-associahedra 的砖多面体几何实现

Geometric realizations of $ν$-associahedra via brick polyhedra

本文利用砖多面体实现了$\ u$-associahedron的几何构造，其边图为$\ u$-Tamari格，并给出了优雅的顶点坐标描述。

Pseudograph associahedra

Given a simple graph G, the graph associahedron KG 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.

球面子词复形的砖多胞形与广义associahedra

Brick polytopes of spherical subword complexes and generalized associahedra

将砖多胞形推广至有限Coxeter群的球面子词复形，为簇复形提供多胞形实现，并给出广义associahedra的顶点描述、Minkowski分解及交换矩阵组合描述。

Coxeter Complexes and Graph-Associahedra

Given a graph G, we construct a simple, convex polytope whose face poset is based on the connected subgraphs of G. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric and combinatorial properties of the complex as well as of the polyhedra are given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford compactification of the real moduli space of curves.

Visibility graphs and deformations of associahedra

The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P. We describe topological properties of this complex and provide realizations based on secondary polytopes. Moreover, using the visibility graph of P, a deformation space of polygons is created which encapsulates substructures of the associahedron.

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.

The Combinatorics of Directed Planar Trees

We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are homeomorphic to balls. We also calculate the number of vertices of the lowest generalized associahedra, giving appropriate generalizations of the Catalan numbers.

图关联体和巢状体的形变锥

Deformation cones of graph associahedra and nestohedra

本文给出了图关联体和巢状体形变锥的刻面描述，推广了次模函数锥参数化，并构造了推广运动学关联体的运动学巢状体。

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 $Θ(m)$ on the diameter of trivially perfect graph associahedra on $m$ edge…

Inequalities between gamma-polynomials of graph-associahedra

We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $γ$-polynomials of their associated graph-associahedra. The partial order is given by relating trees that can be obtained from one another by operations called tree shifts. We also show that tree shifts lower the $γ$-polynomials of graphs that are not trees, as do the flossing moves of Babson and Reiner.

Algebraic structures on graph associahedra

M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph $Γ$, in the context of configuration spaces on the Hilbert plane. To any finite simple graph $Γ$ they associated a finite partially ordered set, whose elements are the tubings of $Γ$ and whose geometric realization is a convex polytope ${\mathcal K}Γ$, the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for simple graph they obtained the standard simplexes. The goal of the present work is to give an \emph{algebraic} descript…

Root systems and graph associahedra

It is known that a connected simple graph $G$ associates a simple polytope $P_G$ called a graph associahedron in Euclidean space. In this paper we show that the set of facet vectors of $P_G$ forms a root system if and only if $G$ is a cycle graph and that the root system is of type A.

Many non-equivalent realizations of the associahedron

Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify the associahedra obtained by these constructions modulo linear equivalence of their normal fans and show, in particular, that the only realization that can be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002) associahedron. For the Hohlweg-Lange associahedra our classification…

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…

Hypergraph Polytopes

We investigate a family of polytopes introduced by E.M.\ Feichtner, A.\ Postnikov and B.\ Sturmfels, which were named nestohedra. The vertices of these polytopes may intuitively be understood as constructions of hypergraphs. Limit cases in this family of polytopes are, on the one end, simplices, and, on the other end, permutohedra. In between, as notable members one finds associahedra and cyclohedra. The polytopes in this family are investigated here both as abstract polytopes and as realized in Euclidean spaces of all finite dimensions. The later realizations are inspired by J.D.\ Stasheff's …

通过脊柱构造的结合多面体

Associahedra via spines

本文通过引入三角剖分的脊柱（对偶树加标记和定向），扩展了结合多面体的经典理解，为Hohlweg-Lange构造提供新视角，并简化了相关证明。

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. A particularly elegant realization of the associahedron, due to S. Shnider and S. Sternberg and popularized by J.-L. Loday, has been generalized in two directions: on the one hand by A. Postnikov to obtain a realization of the graph associahedra of M. Carr and S. Devadoss, and on the other hand by C. Hohlweg and C. Lange to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and…

Permutahedra and generalized associahedra

Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the con…

四类associahedra模型之间的等价性

Equivalence Between Four Models of Associahedra

本文研究了四类associahedra模型之间的等价性，揭示了Stasheff associahedra与Loday构造的同构关系，并探讨了multiplihedra和graph cubeahedra之间的联系。

Realizing the associahedron: Mysteries and questions

There are many open problems and some mysteries connected to the realizations of the associahedra as convex polytopes. In this note, we describe three -- concerning special realizations with the vertices on a sphere, the space of all possible realizations, and possible realizations of the multiassociahedron.

D型associahedra的直径与非离开面性质

The diameter of type D associahedra and the non-leaving-face property

本文证明D型associahedron的图直径为2n-2，并证明所有BCD型associahedra满足非离开面性质，同时给出反例说明该性质并非普遍成立。

Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi, then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi) equals the nested set complex of A. In addition, we prove that for any orientable matroid M, one can find |mu(M)| different reorientations of …

图 associahedra 的谱与扩张

On the spectrum and expansion of graph associahedra

本文对图 associahedra 进行谱分析，给出其第二大特征值下界，利用公平划分分析 stellohedron 的谱，证明该多面体在特定区间存在特征值、有高重数特征值及两个小特征值。

Upper and lower bound theorems for graph-associahedra

From the paper of the first author it follows that upper and lower bounds for $γ$-vector of a simple polytope imply the bounds for its $g$-,$h$- and $f$-vectors. In the paper of the second author it was obtained unimprovable upper and lower bounds for $γ$-vectors of flag nestohedra, particularly Gal's conjecture was proved for this case. In the present paper we obtain unimprovable upper and lower bounds for $γ$-vectors (consequently, for $g$-,$h$- and $f$-vectors) of graph-associahedra and some its important subclasses. We use the constructions that for an $(n-1)$-dimensional graph-associahedr…

图结合多面体的小覆盖与闭链的实现

Small covers of graph-associahedra and realization of cycles

本文研究图结合多面体小覆盖的普适实现性质，证明连通图对应的小覆盖或其二重定向覆盖为 URC-流形，并给出 Stasheff 结合多面体的更简单例子。

The diameter of caterpillar associahedra

The caterpillar associahedron $\mathcal{A}(G)$ is a polytope arising from the rotation graph of search trees on a caterpillar tree $G$, generalizing the rotation graph of binary search trees (BSTs) and thus the conventional associahedron. We show that the diameter of $\mathcal{A}(G)$ is $Θ(n + m \cdot (H+1))$, where $n$ is the number of vertices, $m$ is the number of leaves, and $H$ is the entropy of the leaf distribution of $G$. Our proofs reveal a strong connection between caterpillar associahedra and searching in BSTs. We prove the lower bound using Wilber's first lower bound for dynamic BS…

On the Diameter of Tree Associahedra

We consider a natural notion of search trees on graphs, which we show is ubiquitous in various areas of discrete mathematics and computer science. Search trees on graphs can be modified by local operations called rotations, which generalize rotations in binary search trees. The rotation graph of search trees on a graph $G$ is the skeleton of a polytope called the graph associahedron of $G$. We consider the case where the graph $G$ is a tree. We construct a family of trees $G$ on $n$ vertices and pairs of search trees on $G$ such that the minimum number of rotations required to transform one se…

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

To any graph $G$ one can associate a toric variety $X(\mathcal{P}G)$, obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of $G$. The polytope of this toric variety is the graph associahedron of $G$, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space $X(\mathcal{P}{G})$ is isomorphic to a Hassett compactification of $M_{0,n}$ 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 …

Three non-equivalent realizations of the associahedron

We review three realizations of the associahedron that arise as secondary polytopes, from cluster algebras, and as Minkowski sums of simplices, and show that under any choice of parameters, the resulting associahedra are affinely non-equivalent.

无环偏序多面体及其商

Acyclic Poset Multiplihedra and their Quotients

研究利用图乘法多面体为无环关联多面体定义并实现新的乘法多面体，还对其商进行了研究，最终得出偏序集序多面体的区间多面体，为结合运算组合学提供了新的多面体相关内容。

Colorful Associahedra and Cyclohedra

Every n-edge colored n-regular graph G naturally gives rise to a simple abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic to G. The paper describes colorful polytope versions of the associahedron and cyclohedron. Like their classical counterparts, the colorful associahedron and cyclohedron encode triangulations and flips, but now with the added feature that the diagonals of the triangulations are colored and adjacency of triangulations requires color preserving flips. The colorful associahedron and cyclohedron are derived as colorful polytopes from the edge colore…

源自图结合多面体的格中的极大链：从Tamari格到弱序

Maximal chains in lattices from graph associahedra: Tamari to the weak order

本文研究了推广弱序和Tamari格的极大管集格的极大链，主要针对在路图与完全图间插值的棒棒糖图，发现其格由满足312-避免推广的排列组成，并在Edelman-Greene双射下将最大长度链关联至部分可移表，进而求出相关对称函数的展开。

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.

图嵌套复形的相容性扇

Compatibility fans for graphical nested complexes

本文受有限型簇复形类比启发，将Fomin-Zelevinsky相容性扇构造推广至图形嵌套复形，定义管间相容度，证明其相容向量支撑完整单纯扇，实现图嵌套复形，并恢复路径情形的加泰罗尼亚多个结合多面体实现。

在图关联多面体和超图多面体上计算距离是FPT的，但在超图多面体上是W[2]-难的

Computing distances is FPT on graph associahedra and W[2]-hard on hypergraphic polytopes

研究在图关联多面体上计算距离的FPT算法，并证明在超图多面体上该问题的W[2]-难性，同时探讨其近似难性。

Faces of Generalized Permutohedra

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and gamma-vectors involving descent statistics. This includes a combinatorial interpretation for gamma-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal's conjecture on nonnegativity of gamma-vectors. We calculate expli…

图运算对图关联多面体的影响

Effect of graph operations on graph associahedra

本研究探讨添加单纯顶点、真/假孪生顶点、删除真孪生诱导子图边集等图运算对旋转图结构的影响，推导色数保持条件，给出直径新下界并算出部分完全二部图旋转图直径精确值。

$P$-associahedra（$P$-结合多面体）

$P$-associahedra

该研究针对偏序集$P$构造了$P$-结合多面体$A(P)$，引入仿射偏序环面多面体并将其实现为圆周上$n$点构型空间的紧化，可得到多种经典多面体作为特例。

有限型簇代数的关联多面体与 $\mathbf{g}$-向量的最小关系

Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors

本文证明了有限型簇代数中网格突变是 $\mathbf{g}$-向量的最小关系，并由此推导出 $\mathbf{g}$-向量扇的多面体实现空间为单纯锥，且可描述为正象限与仿射空间的交集；同时推广到 gentle 代数的非亲吻扇，并证明了相关范畴的代数结果。

Isometry classes of generalized associahedra

Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclidean space with simple roots $Δ=\{\a_s | s\in S\}$ of the same length and fundamental weights $Δ^*=\{v_s | s\in S\}$. We set $M(e)=\sum_{s\in S}κ_s v_s$, $κ_s>0$, and for $w\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\{M(w) | w\in W\}$. Given a Coxeter element $c\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained …

图associahedron上组合最短路径求解的困难性

Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra

该研究证明图associahedron顶点间组合最短路径求解为NP难，解决了相关开放问题，并推论多拟阵基多胞体的该类路径无法多项式时间求解，揭示拟阵与多拟阵的计算差异。

Coloring Planar Graphs via Colored Paths in the Associahedra

Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class \(\mathfrak H\) of triangulations of the 2-sphere that have a Hamiltonian circuit. This has been used by Whitney and others from 1936 to the present to find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has been activity to try to use some of these reformuations to find a shorter proof of the 4CT. Every triangulation in \(\mathfrak H\) has a dual graph that is a union of two binary trees with the same number …

Reconnectad结构

Reconnectads

本文引入reconnectad这一类operad新结构，以有限简单图为输入、通过子图重连补集实现合成，以Carr和Devadoss图结合体复曲面簇为典型例子，发展其一般理论并用于研究由同调群组装的“奇妙reconnectad”。

alt ν- associahedron 的标准实现

A canonical realization of the alt $ν$-associahedron

本文研究了alt ν-Tamari格子的几何结构，提出了一种称为alt ν- associahedron的polytopal complex，并通过面积下的lattice path构造了标准实现，从而在经典associahedron的情况下恢复了Loday的实现。

Amplitubes：图宇宙体

Amplitubes: Graph Cosmohedra

本文推广宇宙体到任意图，定义宇宙学振幅管，通过对图宇宙体顶点求和计算波函数，并给出显式嵌入。

On realization graphs of degree sequences

Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.

On fractional realizations of graph degree sequences

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope. We describe properties of the polytope vertices and characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.

Cliques in realization graphs

The realization graph $\mathcal{G}(d)$ of a degree sequence $d$ is the graph whose vertices are labeled realizations of $d$, where edges join realizations that differ by swapping a single pair of edges. Barrus [On realization graphs of degree sequences, Discrete Mathematics, vol. 339 (2016), no. 8, pp. 2146-2152] characterized $d$ for which $\mathcal{G}(d)$ is triangle-free. Here, for any $n \geq 4$, we describe a structure in realizations of $d$ that exactly determines whether $G(d)$ has a clique of size $n$. As a consequence we determine the degree sequences $d$ for which $\mathcal{G}(d)$ is…

Realization of distance matrices by graphs of genus 1

Given a distance matrix $D$, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of $D$ by a weighted graph. To this end, we first give a general result on realization by $n-$cycles and successively we mainly focus on graphs of genus 1, presenting an algorithm which determines when a distance matrix is realizable by such a kind of graph, and then, shows how to construct it.

Graphs with Flexible Labelings allowing Injective Realizations

We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinator…

Topological realizations and fundamental groups of higher-rank graphs

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor, and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely gen…