Related papers: Combinatorics of flag simplicial 3-polytopes
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…
Abstract polytopes are a combinatorial generalization of convex and skeletal polytopes. Counting how many flag orbits a polytope has under its automorphism group is a way of measuring how symmetric it is. Polytopes with one flag orbit are…
We describe a natural geometric relationship between matroids and underlying flag matroids by relating the geometry of the greedy algorithm to monotone path polytopes. This perspective allows us to generalize the construction of underlying…
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the…
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular,…
The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph of the symmetric group S_p and then construct a…
In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete…
In this note, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by using only one edge contraction? We show that the problem is polynomial-time solvable on $P_3+kP_2$-free graphs for…
Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs.…
Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a $k$-connected graph is {\em contractible} if its contraction does not result in a graph of lower…
We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on…
We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold.…
We prove that the family of facets of a pure simplicial complex of dimension up to three satisfies the Erd\H{o}s-Ko-Rado property whenever it is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and…
For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a…
The problem of finding a triangulation of a convex three-dimensional polytope with few tetrahedra is proved to be NP-hard. We discuss other related complexity results.
We study dismantlability in graphs. In order to compare this notion to similar operations in posets (partially ordered sets) or in simplicial complexes, we prove that a graph G dismants on a subgraph H if and only if H is a strong…
Given any polytope $P$ and any generic linear functional ${\bf c} $, one obtains a directed graph $G(P,{\bf c})$ from the 1-skeleton of $P$ by orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. For $P$ a simple…
The convex hull $\psi_{n,n}$ of certain $(n!)^2$ tensors was considered recently in connection with graph isomorphism. We consider the convex hull $\psi_n$ of the $n!$ diagonals among these tensors. We show: 1. The polytope $\psi_n$ is a…
Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we…
First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and…