A realization of graph associahedra

Abstract

Given any finite graph G, we offer a simple realization of the graph-associahedron PG using integer coordinates.2000 Mathematics Subject Classification. Primary 52B11.

Convexifying Monotone Polygons while Maintaining Internal Visibility

“…Devadoss et al [8] showed how to convexify star-shaped polygons by moving all the vertices in the polygon simultaneously. We believe that our techniques should extend to star-shaped polygons, but so far we have not been able to do it.…”

“…On the other hand, we give an example that shows that the answer to Question (1) is negative, even for monotone polygons. For recent results on this topic, see also [8].…”

“…Our starting point is the following question posed by Satyan L. Devadoss in the Open Problem Session at CCCG 2008 [7,8]:…”

“…Two vertices of P are visible if the open line segment joining them is contained in the interior of P . In this paper we study the following questions posed in [7,8]: (1) Is it true that every non-convex simple polygon has a vertex that can be continuously moved such that during the process no vertex-vertex visibility is lost and some vertex-vertex visibility is gained? (2) Can every simple polygon be convexified by continuously moving only one vertex at a time without losing any internal vertex-vertex visibility during the process?…”

Visibility-preserving convexifications using single-vertex moves

“…(equivalently: does every polygon admit a visibility-preserving convexification?) Devadoss [5] also raised a stronger version of this question: (2) does there exist a visibility-preserving convexification in which at most one vertex is moved at a time (that is, by singlevertex moves)? Partial answers to (1) and (2) were given in [2] and [5].…”