By van Leeuwen E.
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Extra info for Optimization and approximation on systems of geometric objects
Problems We consider various optimization problems on graphs that are relevant to geometric intersection graph models and specifically to (unit) disk graphs models of wireless communication networks. 1 Let G be a graph. A set S ⊆ V (G) is an independent set if there are no u, v ∈ S such that (u, v) ∈ E(G). A set S ⊆ V (G) is a vertex cover if for each (u, v) ∈ E(G) it holds that u ∈ S or v ∈ S. Observe that an independent set is the complement of a vertex cover (and vice versa) . Furthermore, we are usually looking for a maximum independent set and a minimum vertex cover.
215] give a polynomial-time algorithm yielding √ a mapping of quality O((log5/2 n) · log log n). This clearly leaves a large gap and a major open question. 2. Disk Graphs and Ball Graphs 23 ball contact graphs. In two dimensions, ball contact graphs are also called disk contact graphs or coin graphs . 1). Hence these graphs are recognizable in linear time . If however the ratio of the radii of the largest and smallest disk is any fixed constant, then the recognition problem becomes NP-hard .
Recognizing whether the boxicity is at most d is NP-complete for any fixed d ≥ 2 [173, 273, 196]. Note that for d = 1, the recognition problem is equal to the problem of recognizing interval graphs, which is in P. Given the above, it is not hard to imagine a natural generalization of unit interval graphs. This leads to intersection graphs of d-dimensional axis-parallel unit cubes, which are d-dimensional axis-parallel boxes with side length equal to one. For d = 2, these are called unit square intersection graphs, or simply unit square graphs.
Optimization and approximation on systems of geometric objects by van Leeuwen E.