In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter quest...

Let P be a simple polygon of n vertices and let S be a set of N points lying in the interior of P . A geodesic disk GD(p, r) with center p and radius r is the set of points in P that have a geodesic distance ≤ r from p (where the geodesic distance is the length of the shortest polygonal path connection that lies in P ). In this paper we present an output sensitive algorithm for finding all N ge...

This article studies the relationship between simple closed geodesics and orientation reversing involutions on one-holed hyperbolic tori. 1. Simple closed geodesics on the one holed torus We consider topological surfaces of signature (g, n) (where g is the underlying genus and n is the number of simple boundary curves) with negative Euler characteristic and endowed with a hyperbolic metric such...

We propose a family of discrete space-time quantum walks capable propagating on any arbitrary triangulations. Moreover, we also extend and generalize the duality principle introduced by Arrighi et al. [Sci. Rep. 9, 10904 (2019)], linking continuous local deformations given triangulation inhomogeneity unitaries that guide walker. proved in formal limit, both space time, this converges to ($1+2$)...

The planar trigonal structure of 1,3,5-trisubstituted benzene, named phenine, has been adopted as basic units polygonal networks to shape nanometer-sized curved organic π-molecules. phenine design allows for concise syntheses large carbonaceous molecules reaching 4 kDa by stitching geodesic lines with aryl coupling reactions. In this Account, the development defective nanocarbon molecules, i.e....