Deformations of associahedra and visibility graphs

Given an arbitrary simple polygon, we construct a polytopal complex analogous to the associahedron based on its convex diagonalizations. This polytopal complex is shown to be contractible, and a geometric realization is provided based on the theory of secondary polytopes. We then reformulate a combinatorial deformation theory in terms of visibility and presents some open problems

On Colourability of Polygon Visibility Graphs

We study the problem of colouring the visibility graphs of polygons. In particular, we provide a polynomial algorithm for 4-colouring of the polygon visibility graphs, and prove that the 6- colourability question is already NP-complete for them

Visibility-monotonic polygon deflation

A deflated polygon is a polygon with no visibility crossings.  We answer a question posed by Devadoss et al. (2012) by presenting a  polygon that cannot be deformed via continuous visibility-decreasing  motion into a deflated polygon.  We show that the  least n for which there exists such an n-gon is seven.  In  order to demonstrate non-deflatability, we use a new combinatorial  structure for polygons, the directed dual, which encodes the  visibility properties of deflated polygons.  We also show that any  two deflated polygons with the same directed dual can be deformed,  one into the other, through a visibility-preserving deformation

Convexity-Increasing Morphs of Planar Graphs

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201

Upward Planar Morphs

We prove that, given two topologically-equivalent upward planar straight-line drawings of an $n$-vertex directed graph $G$, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of $O(1)$ morphing steps if $G$ is a reduced planar $st$-graph, $O(n)$ morphing steps if $G$ is a planar $st$-graph, $O(n)$ morphing steps if $G$ is a reduced upward planar graph, and $O(n^2)$ morphing steps if $G$ is a general upward planar graph. Further, we show that $\Omega(n)$ morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an $n$-vertex path.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018) The current version is the extended on

Convexifying Polygons Without Losing Visibilities

We show that any simple n-vertex polygon can be made convex, without losing internal visibilities between vertices, using n moves. Each move translates a vertex of the current polygon along an edge to a neighbouring vertex. In general, a vertex of the current polygon represents a set of vertices of the original polygon that have become co-incident. We also show how to modify the method so that vertices become very close but not co-incident, in which case we need O(n²) moves, where each move translates a single vertex. The proof involves a new visibility property of polygons, namely that every simple polygon has a visibilityincreasing edge where, as a point travels from one endpoint of the edge to the other, the visibility region of the point increases