Consider an art gallery formed by a polygon on n vertices with m pairs of vertices joined by interior diagonals, the interior walls. Each interior wall has an arbitrarily placed, arbitrarily small doorway. We will show that the minimum number of guards that suffice to guard all art galleries with n vertices and m interior walls is min{b(2n−3)/3c, b(2n+m−2)/4c, b(2m+n)/3c}. If we restrict oursel...

We discovered the new tiling patterns each of which is composed of a single kind of convex pentagon. Moreover, we propose a new concept of classification method of the tessellating convex pentagons, which is the only unsolved case among the corresponding tessellating convex polygon problems.

We give infinite lists of translations surfaces with no convex presentations. We classify the surfaces in the stratum H(2) which do not have convex presentations, as well as those with no strictly convex presentations. We show that in H(1, 1), all surfaces in the eigenform loci E4, E9 or E16 have no strictly convex presentation, and that the list of surfaces with no convex presentations in H(1,...

Strong visibility is a generalization of standard visibility, defined with respect to a fixed set of line orientations. We investigate computational properties of this generalized visibility, as well as the related notion of strong convexity. In particular, we describe algorithms for the following tasks: 1. Testing the strong visibility of two points in a polygon and the strong convexity of a p...

In this paper we address several variants of the polygon an-nulus placement problem: given an input polygon P and a set S of points, nd an optimal placement of P that maximizes the number of points in S that fall in a certain annulus region deened by P and some ooset distance > 0. We address the following variants of the problem: placement of a convex polygon as well as a simple polygon; placem...

We study the relationship between the interior angles of a convex polygon and the lengths of external watchman routes.