Summary: Weighted Skeletons and FixedShare Decomposition #
Institute for Theoretical Computer Science
University of Technology, Graz, Austria
We introduce the concept of weighted skeleton of a polygon and present various decomposition and
optimality results for this skeletal structure when the underlying polygon is convex.
Polygon decomposition is a major issue in computational geometry. Its relevance stems from breaking com
plex shapes (modeled by polygons) into subpolygons that are easier to manipulate, and from subdividing
areas of interest into parts that satisfy certain containment requirements and/or optimality properties. We
refer to  for a nice survey on this topic. In particular, a rich literature exists on decomposition into
convex polygons. Convex decompositions are most natural in some sense. They have many applications
and can be computed efficiently; see e.g. [7, 14, 16].
In this paper, we focus on the problem of decomposing a convex polygon such that predefined constraints
are met. More specifically, the goal is to partition a given convex ngon P into n convex parts, each part
being based on a single side of P and containing a specified 'share' of P . The share may relate, for
example, to the spanned area, to the number of contained points from a given point set, or to the total edge
length covered from a given set of curves. Possible applications of such fixedshare decompositions include