We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a "vertex-faithful" polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime. We also construct the smallest regular polyhedra with a prime squared number of vertices.
In this article we study surface subgroups of groups acting simply transitively on vertex sets of certain hyperbolic triangular buildings. Kangaslampi and Vdovina have constructed and classified all groups acting simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. These groups gave the first examples of cocompact lattices acting simply transitively on vertices of hyperbolic triangular Kac-Moody buildings that are not right-angled. The hyperbolic buildings arise as universal covers of polyhedra glued together from with 15 triangular face…
A sequence $σ$ of $p$ non-negative integers is unigraphic if it is the degree sequence of exactly one graph, up to isomorphism. A polyhedral graph is a $3$-connected, planar graph. We investigate which sequences are unigraphic with respect to the class of polyhedral graphs, meaning that they admit exactly one realisation as a polyhedron. We focus on the case of sequences with largest entry $p-2$. We give a classification of polyhedral unigraphic sequences starting with $p-2,p-2$, as well as those starting with $p-2$ and containing exactly one $3$. Moreover, we characterise the unigraphic seque…
An unfolding of a polyhedron along its edges is called a vertex unfolding if adjacent faces are allowed to be connected at not only an edge but also a vertex. Demaine et al showed that every triangulated polyhedron has a vertex unfolding. We extend this result to a tight polyhedron, where a polyhedron is tight if all non-triangular faces are mutually non-adjacent parallelograms.
This paper focuses on vertices of the master corner polyhedra $P(G,g_0),$ the core of the group-theoretical approach to integer linear programming. We introduce two combinatorial operations that transform each vertex of $P(G,g_0)$ to adjacent ones. This implies that for any $P(G,g_0),$ there exists a subset of basic vertices, we call them support vertices, from which all others can be built. The class of support vertices is proved to be invariant under the automorphism group of $G,$ so this basis can be further reduced to a subset of pairwise non-equivalent support vertices. Among other result…
Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction is applied to the forty-eight regular skeletal polyhedra (Grunbaum-Dress polyhedra) in ordinary space, new highly symmetric skeletal polyhedra arise as "truncations" of the original polyhedra. These Wythoffians are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes. The present paper describes the bluep…
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particu…
Given vertex valencies admissible for a self-dual polyhedral graph, we describe an algorithm to explicitly construct such a polyhedron. Inputting in the algorithm permutations of the degree sequence can give rise to non-isomorphic graphs. As an application, we find as a function of $n\geq 3$ the minimal number of vertices for a self-dual polyhedron with at least one vertex of degree $i$ for each $3\leq i\leq n$, and construct such polyhedra. Moreover, we find a construction for non-self-dual polyhedral graphs of minimal order with at least one vertex of degree $i$ and at least one $i$-gonal fa…
When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a classification of polyhedra.
Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their evolutes is obtained. Also are considered higher-dimensional analogs of the four-vertex theorem for regular and shellable triangulations.
Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern--Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.
Over the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorithms for high-dimensional polyhedra given by inequalities. It is a famous and long standing open question in polyhedral theory and computational geometry whether the vertices of a polytope (bounded polyhedron), described by a set of linear constraints, can be enumerated in total polynomial time. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra $P = \{x : A…
Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and $S$ can be extended to a $k$-link between $v$ and $S$ such that the endvertices of the $(k-1)$-link are also the endvertices of the $k$-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets $S_1$ and $S_2$ are $k$-linked, then a $t$-link ($t< k$) between two disjoint sets $S_1$ and $S_2$ can be extended to a $k$-l…
We give coordinate-minimal geometric realizations in general position of all 865 vertex-minimal triangulations of the orientable surface of genus 2 in the 4x4x4-cube.
In the article, a series of neigbourly polyhedra is constructed. They have $N=2d+4$ vertices and are embedded in $\mathbb R^{2d}$. Their (affine) Gale diagrams in $\mathbb R^2$ have $d+3$ black points that form a convex polygon. These Gale diagams can be enumerated using 3-trees (trees with some additional structure). Given $d$ and $m$, each of the constructed polyhedra in $\mathbb R^{2d}$ has a fixed number of faces of dimension $m$ that contain a vertex $A$. (This number depends on $d$ and $m$ does not depend on the polyhedron and the vertex $A$).
We give coordinate-minimal geometric realizations in general position for 17 of the 20 vertex-minimal triangulations of the orientable surface of genus 3 in the 5x5x5-cube.
We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary 3-space. This polyhedron was originally discovered by Grunbaum in 1999, but was recently rediscovered by Brehm and Wills. We establish isomorphism of the Grunbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grunbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g > 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combinatorially regular. We also contribute …
We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. T…
The problem of counting the number of waves arriving at the vertex of a polyhedron is motivated by physics. In the article it was solved for the case of Platonic solid using three nontrivial results from number theory. This growth turns out to be subexponential. Also we prove a subexponential upper bound for all polyhedra with rational total angles at vertices.
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is "regular" if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal co…
This paper deals with triangulations of the 2-torus with the vertex labeled general octahedral graph $O_4$ which is isomorphic to the complete four-partite graph $K_{2,2,2,2}$; it is known that there exist precisely twelve such triangulations. We find all the 12 triangulations in a Schlegel diagram of the hyperoctahedron and realize all of them geometrically with the same 1-skeleton in 3-space. In particular, we identify two geometric polyhedral tori (both without self-intersections) with the same 1-skeleton in 3-space, but without a single common face, or in other words their intersection (as…
The notion of a competition graph was introduced by J. E. Cohen in 1968. The competition graph C(D) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many new isolated vertices is the competition graph of some acyclic digraph. In 1978, F. S. Roberts defined the competition number k(G) of a graph G as the minimum number of such isolated vertices. In general, it is hard to compute th…
A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron correspon…
We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which c…
