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Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…
A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A…
A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope.…
Given a tree $T$, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in $T$. These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics,…
The cut polytope of a graph $G$ is the convex hull of the indicator vectors of all cuts in $G$ and is closely related to the MaxCut problem. We give the facet-description of cut polytopes of $K_{3,3}$-minor-free graphs and introduce an…
We consider the convex hull $P_{\varphi}(G)$ of all satisfying assignments of a given MSO formula $\varphi$ on a given graph $G$. We show that there exists an extended formulation of the polytope $P_{\varphi}(G)$ that can be described by…
We give a decomposition of the suspension of a polyhedral join in terms of the polyhedral smash product of the suspension of the family of pairs, and study some cases in which the formula can be desuspended, particularly for polyhedral…
For a lattice polytope $P$, the rank of $P$ is defined by $F-(\dim P+1)$, where $F$ is the number of facets of $P$. In this paper, we study matroid polytopes with small rank. More precisely, we characterize matroid independence polytopes…
The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed…
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs…
A graph class is monotone if it is closed under taking subgraphs. It is known that a monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a so-called tripod, that is, a…
A polytope is called indecomposable if it cannot be expressed nontrivially as a Minkowski sum of other polytopes. Since Gale introduced the concept in 1954, several increasingly strong criteria have been developed to characterize…
The symmetric edge polytope (SEP) of a (finite, undirected) graph is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. SEPs have been studied extensively in the past twenty years. Recently,…
Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence…
A biconvex polytope is a classical and tropical convex hull of finitely many points. Given a biconvex polytope, for each vertex of it we construct a directed bigraph and a gammoid so that the collection of base polytopes of those gammoids…
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…
The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-level matroids generalize series-parallel graphs, which have been already…
We give new characterizations for the class of uniformly dense matroids and study applications of these characterizations to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope…
We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a…
We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising…