Visibility Monotonic Polygon Deflation

A deflated polygon is a polygon with no visibility crossings. We answer a question posed by Devadoss et al. () 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.  Introduction Much work has been done on visibilities of polygons [, ] as well as on their convexification, including work on convexification through continuous motions []. Devadoss et al. [] combine these two areas in asking the following two questions: () Can every polygon be convexified through a deformation in which visibilities monotonically increase? () Can every polygon be deflated (i.e. lose all its visibility crossings) through a deformation in which visibilities monotonically decrease? The first of these questions was answered in the affirmative at CCCG  by Aichholzer et al. []. In this paper, we resolve the second question in the negative by presenting a non-deflatable polygon, shown in Figure a. While it is possible to use ad hoc arguments to demonstrate the non-deflatability of this polygon, we develop a combinatorial structure, the directed dual, that allows us to prove non-deflatability for this and other examples using only combinatorial arguments. We also show that seven is the least n for which there exists a non-deflatable n-gon in general position. As a byproduct of developing the directed dual, we obtain the following additional results: () The vertex-edge visibility graph of a deflated polygon is completely determined by its directed dual; and () any deflated polygon may be monotonically deformed into any other deflated polygon having the same directed dual.  Preliminaries We begin by presenting some definitions. Here and throughout the paper, unless qualified otherwise, we take polygon to mean simple polygon on the plane. A triangulation, T , of a polygon, P , with vertex set V is a partition of P into triangles with vertices in V . The edges of T are the edges of these triangles and we call such an edge ∗School of Computer Science, Carleton University, {jit, vida, nhoda, morin}@scs.carleton.ca  ar X iv :1 20 6. 19 82 v1 [ cs .C G ] 9 J un 2 01 2 a polygon edge if it belongs to the polygon or, else, a diagonal. A triangle of T with exactly one diagonal edge is an ear and the helix of an ear is its vertex not incident to any other triangle of T . Let w and uv be a vertex and edge, respectively, of a polygon, P , such that u and v are seen in that order in a counter-clockwise walk along the boundary of P . Then uv is facing w if (u,v,w) is a left turn. Two vertices or a vertex and an edge of a polygon are visible or see each other if there exists a closed line segment contained inside the closed polygon joining them. If such a segment exists that intersects some other line segment then they are visible through the latter segment. We say that a polygon is in general position if the open line segment joining any of its visible pairs of vertices is contained in the open polygon.