Related papers: Steinitz Theorems for Orthogonal Polyhedra

A maniplex of rank n is a connected, n-valent, edge-coloured graph that generalises abstract polytopes and maps. If the automorphism group of a maniplex M partitions the vertex-set of M into k distinct orbits, we say that M is a k-orbit…

In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also…

We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed…

In this paper, we provide a characterization of uniquely representable two-directional orthogonal ray graphs, which are defined as the intersection graphs of rightward and downward rays. The collection of these rays is called a…

An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in…

Let P = k[x_1, ..., x_n] be the polynomial ring in n variables. A homogeneous ideal I of P generated in degree d is called Gotzmann if it has the smallest possible Hilbert function out of all homogeneous ideals with the same dimension in…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as…

Let $P$ be a polygon with $r>0$ reflex vertices and possibly with holes and islands. A subsuming polygon of $P$ is a polygon $P'$ such that $P \subseteq P'$, each connected component $R$ of $P$ is a subset of a distinct connected component…

A convex combination mapping of a planar graph is a plane mapping in which the external vertices are mapped to the corners of a convex polygon and every internal vertex is a proper weighted average of its neighbours. If a planar graph is…

Toeplitz's Square Peg Problem asks whether every continuous simple closed curve in the plane contains the four vertices of a square. It has been proved for various classes of sufficiently smooth curves, some of which are dense, none of…

The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the…

An integral convex polytope ${\mathcal P}$ is said to be Gorenstein if its toric ring $K[{\mathcal P}]$ is normal and Gorenstein. In this paper, Gorenstein cut polytopes of graphs are characterized explicitly. First, we prove that…

In 2020, Cameron et al. introduced the restricted numerical range of a digraph (directed graph) as a tool for characterizing digraphs and studying their algebraic connectivity. In particular, digraphs with a restricted numerical range of a…

If all tiles in a tiling are congruent, the tiling is called monohedral. Tiling by convex polygons is called edge-to-edge if any two convex polygons are either disjoint or share one vertex or one entire edge in common. In this paper, we…

Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from…

The dual of a polyhedron is a polyhedron -- or in graph theoretical terms: the dual of a 3-connected plane graph is a 3-connected plane graph. Astonishingly, except for sufficiently large facewidth, not much is known about the connectivity…

Using the orthogonal connectedness, we introduce the notion of orthogonal decomposability of convex polytopes and study it in the case of Platonic and Archimedean solids. While doing so, we also encounter polytopes which are not…

The shape of crystalline nanoparticles (NP) can often be described by polyhedra with flat facet surfaces. Thus, structural studies of polyhedral bodies can help to describe geometric details of NPs. Here we consider compact polyhedra of…

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the…