Compact Visibility Representation of Plane Graphs

The visibility representation (VR for short) is a classical representation of plane graphs. It has various applications and has been extensively studied. A main focus of the study is to minimize the size of the VR. It is known that there exists a plane graph $G$ with $n$ vertices where any VR of $G$ requires a grid of size at least (2/3)n x((4/3)n-3) (width x height). For upper bounds, it is known that every plane graph has a VR with grid size at most (2/3)n x (2n-5), and a VR with grid size at most (n-1) x (4/3)n. It has been an open problem to find a VR with both height and width simultaneously bounded away from the trivial upper bounds (namely with size at most c_h n x c_w n with c_h < 1 and c_w < 2$). In this paper, we provide the first VR construction with this property. We prove that every plane graph of n vertices has a VR with height <= max{23/24 n + 2 Ceil(sqrt(n))+4, 11/12 n + 13} and width <= 23/12 n. The area (height x width) of our VR is larger than the area of some of previous results. However, bounding one dimension of the VR only requires finding a good st-orientation or a good dual s^*t^*-orientation of G. On the other hand, to bound both dimensions of VR simultaneously, one must find a good$st$-orientation and a good dual s^*t^*-orientation at the same time, and thus is far more challenging. Since st-orientation is a very useful concept in other applications, this result may be of independent interests

A Quantitative Steinitz Theorem for Plane Triangulations

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation $G$ with $n$ vertices can be embedded in $\mathbb{R}^2$ in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a $4n^3 \times 8n^5 \times \zeta(n)$ integer grid, where $\zeta(n) \leq (500 n^8)^{\tau(G)}$ and $\tau(G)$ denotes the shedding diameter of $G$, a quantity defined in the paper.Comment: 25 pages, 6 postscript figure

Morphing planar triangulations

A morph between two drawings of the same graph can be thought of as a continuous deformation between the two given drawings. A morph is linear if every vertex moves along a straight line segment from its initial position to its final position. In this thesis we study algorithms for morphing, in which the morphs are given by sequences of linear morphing steps. In 1944, Cairns proved that it is possible to morph between any two planar drawings of a planar triangulation while preserving planarity during the morph. However this morph may require exponentially many steps. It was not until 2013 that Alamdari et al. proved that the morphing problem for planar triangulations can be solved using polynomially many steps. In 1990 it was shown by Schnyder that using special drawings that we call Schnyder drawings it is possible to draw a planar graph on a O(n)×O(n) grid, and moreover such drawings can be found in O(n) time (here n denotes the number of vertices of the graph). It still remains unknown whether there is an efficient algorithm for morphing in which all drawings are on a polynomially sized grid. In this thesis we give two different new solutions to the morphing problem for planar triangulations. Our first solution gives a strengthening of the result of Alamdari et al. where each step is a unidirectional morph. This also leads to a simpler proof of their result. Our second morphing algorithm finds a planar morph consisting of O(n²) steps between any two Schnyder drawings while remaining in an O(n)×O(n) grid. However, there are drawings of planar triangulations which are not Schnyder drawings, and for these drawings we show that a unidirectional morph consisting of O(n) steps that ends at a Schnyder drawing can be found. We conclude this work by showing that the basic steps from our morphs can be implemented using a Schnyder wood and weight shifts on the set of interior faces