Related papers: Hypergraph Polytopes
It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$-polytopes with at most $d-2$ nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2…
We consider families of simple polytopes $P$ and simplicial complexes $K$ well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics…
Symmetric edge polytopes, a.k.a. PV-type adjacency polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In…
Beck et. al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters, perfect matching polytopes of torus graphs are Gorenstein. In this paper, we complement their result,…
We study graph complexes related to configuration spaces and diffeomorphism groups of highly connected manifolds of odd dimension. In particular we compute the cohomology in the "high genus" limit. This paper is a continuation of previous…
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…
An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and…
We identify a family of $O(|E(G)|^2)$ nontrivial facets of the connected matching polytope of a graph $G$, that is, the convex hull of incidence vectors of matchings in $G$ whose covered vertices induce a connected subgraph. Accompanying…
Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and…
We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used…
The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to…
We prove the combinatorial property of shortness for freehedra. Note that associahedra, a sibling family of polytopes, are not short.
Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…
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…
We prove that the inclusion from oriented graph complex into graph complex with at least one source is a quasi-isomorphism, showing that homology of the "sourced" graph complex is also equal to the homology of standard Kontsevich's graph…
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…
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…
Matroids give rise to several natural constructions of polytopes. Inspired by this, we examine polytopes that arise from the signed circuits of an oriented matroid. We give the dimensions of these polytopes arising from graphical oriented…
We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P. We devise a combinatorial method…
In their seminal paper introducing the theory of random graphs, Erd\H{o}s and R\'{e}nyi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this…