Related papers: Rectangular Duals on the Cylinder and the Torus
We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: (1) A toroidal graph embedding $\Gamma$ is positive equilibrium if it is possible to place positive weights on the edges, such that the weighted edge vectors…
A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of $G$ to a sequence of horizontal…
In 2009 Chmutov introduced the idea of partial duality for embeddings of graphs in surfaces. We discuss some alternative descriptions of partial duality, which demonstrate the symmetry between vertices and faces. One is in terms of band…
We consider the Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families $\calR$ where $R_1 \setminus R_2$…
We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of…
We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image…
Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more…
(0,2) gauged linear sigma models with torsion, corresponding to principal torus bundles over warped CY bases, provide a useful framework for getting exact statements about perturbative dualities in the presence of fluxes. In this context we…
A self-dual map $G$ is said to be \emph{antipodally self-dual} if the dual map $G^*$ is antipodal embedded in $\mathbb{S}^2$ with respect to $G$. In this paper, we investigate necessary and/or sufficient conditions for a map to be…
Any planar graph has a crossing-free straight-line drawing in the plane. A simultaneous geometric embedding of two n-vertex graphs is a straight-line drawing of both graphs on a common set of n points, such that the edges withing each…
The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and delta-matroids. There are well-known connections…
We consider cylinders in ${\cal H}^2\times R$ (see definitions in the introduction) and prove that a complete and connected surface in ${\cal H}^2\times R$ with the vanishing of the Gauss and extrinsic curvatures is a cylinder.
Two planar graphs G1 and G2 sharing some vertices and edges are `simultaneously planar' if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether…
It is shown that each quadrangulation of the 2-torus by the Cartesian product of two cycles can be geometrically realized in (Euclidean) 4-space without hidden symmetries---that is, so that each combinatorial cellular automorphism of the…
Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots…
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…
Every simple quadrangulation of the sphere is generated by a graph called a pseudo-double wheel with two local expansions (Brinkmann et al. "Generation of simple quadrangulations of the sphere." Discrete Math., Vol. 305, No. 1-3, pp. 33-54,…
We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our…
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While…
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…