By van Leeuwen E.
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Additional resources for Optimization and approximation on systems of geometric objects
The mixing of both object types is not considered here. Hence we restrict to cs ∈ int(s). We can easily determine which objects are scalable and which are not. An object s is said to be strongly star-shaped if there is a point ts ∈ int(s) such that for any point p ∈ s the straight line segment ts p is contained in s, but does not contain any point of bd(s), except possibly p. 3 An object s is scalable if and only if it is strongly starshaped. Proof: Suppose that s is scalable and has scaling point cs .
6, we think of -separated as being a slightly more general notion, since the property of being -separated is invariant under a scaling of the space. 32 Chapter 4. Geometric Intersection Graphs and Their Representation We now show that any intersection graph of scalable objects has an separated representation. 7 For a family A of closed scalable objects, any A-intersection graph has an -separated representation for some > 0. Proof: Let G be an A-intersection graph and S any representation of G. We prove that S can be turned into an -separated representation of G.
5 can be proved for intersection graphs of other scalable objects. In particular, we conjecture that similar techniques apply to intersection graphs of (unit) regular hexagons. Finally, observe that for the results in this section it does not matter if the disks or squares are open or closed. 2 From Separation to Representation The above theorems were quite specific to the object type. We can prove that the converse holds in a more general setting. In the following, let zs denote the size of an object s.