We present two algorithms for packing two largest disks in a polygon. The first algorithm locates two disks in a simple polygon in time O(n log2 n) improving the best previous deterministic result (Bespamyatnikh, 1999) by a factor of logn. The second algorithm finds two disks in a convex polygon such that the disks are separated by a diagonal of the polygon. It runs in time O(n log2 n) improvin...

Let Σ = { S1 , . . . , Sn } be a finite set of disjoint line segments in the plane. We conjecture that its visibility graph, Vis(Σ), is hamiltonian. In fact, we make the stronger conjecture that Vis(Σ) has a hamiltonian cycle whose embedded version is a simple polygon (i.e., its boundary edges are non-crossing visibility segments). We call such a simple polygon a spanning polygon of Σ. Existenc...

Let S be used to denote a nite set of planar geometric objects. Deene a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have som...

In the Art Gallery problem a polygon is given and the goal is to place as few guards as possible so that the entire area of the polygon is covered. We address a closely related problem: how to place a fixed number of guards on the vertices or the edges of a simple polygon so that the total guarded area inside the polygon is maximized. Recall that an optimization problem is called APX-hard, if t...

The relative convex hull, or the minimum-perimeter polygon (MPP) of a simple polygon A, contained in a second polygon B, is a unique polygon in the set of nested polygons between A and B. The computation of the minimum-length polygon (MLP), as a special case for isothetic polygons A and B, is useful for various applications in image analysis and robotics. The paper discusses the first recursive...