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

Which nestohedra are removahedra?

A removahedron is a polytope obtained by deleting inequalities from the facet description of the classical permutahedron. Relevant examples range from the associahedra to the permutahedron itself, which raises the natural question to characterize which nestohedra can be realized as removahedra. In this note, we show that the nested complex of any connected building set closed under intersection can be realized as a removahedron. We present two different complementary proofs: one based on the building trees and the nested fan, and the other based on Minkowski sums of dilated faces of the standard simplex. In general, this closure condition is sufficient but not necessary to obtain removahedra. However, we show that it is also necessary to obtain removahedra from graphical building sets, and that it is equivalent to the corresponding graph being chordful (i.e. any cycle induces a clique).Comment: 13 pages, 4 figures; Version 2: new Remark 2

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

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 $\Delta$-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 of graphs. We then define hypercube-graph associahedra as a special case. Hypercube-graph associahedra are defined by tubes and tubings on a graph with a matching of dashed edges, with tubes and tubings avoiding those dashed edges. These simple rules make hypercube-graph tubings a simple and intuitive extension of classical graph tubings. We explore properties of $\Delta$-nested complexes and P-nestohedra, and use these results to explore properties of hypercube-graph associahedra, including their facets and faces, as well as their normal fans and Minkowski sum decompositions. We use these properties to develop general methods of enumerating $f$-polynomials of families of hypercube-graph associahedra. Several of these hypercube-graphs correspond to previously-studied polyhedra, such as cubeahedra, the halohedron, the type $A_n$ linear $c$-cluster associahedron, and the type $A_n$ linear $c$-cluster biassociahedron. We provide enumerations for these polyhedra and others.Comment: PhD Thesis of Jordan Almeter, 2022. https://repository.lib.ncsu.edu/handle/1840.20/3992

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

Signed tree associahedra

Abstract. 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. Résumé. 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 realisations 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